?A cubic function is a polynomial of degree 3; that is, it has the form , where a 0.(a)

Explain the difference between an absolute minimum and a local minimum.

Suppose f is a continuous function defined on a closed interval [a,b]. (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for f? (b) What steps would you take to find those maximum and minimum values?

3–4 For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum. 3.

3–4 For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum. 4.

5–6 Use the graph to state the absolute and local maximum and minimum values of the function. 5.

5–6 Use the graph to state the absolute and local maximum and minimum values of the function. 6.

7–10 Sketch the graph of a function that is continuous on [1, 5] and has the given properties. 7. Absolute minimum at 2, absolute maximum at 3, local minimum at 4

7–10 Sketch the graph of a function that is continuous on [1, 5] and has the given properties. 8. Absolute minimum at 1, absolute maximum at 5, local maximum at 2, local minimum at 4

7–10 Sketch the graph of a function that is continuous on [1, 5] and has the given properties. 9. Absolute maximum at 5, absolute minimum at 2, local maximum at 3, local minima at 2 and 4

7–10 Sketch the graph of a function that is continuous on [1, 5] and has the given properties. 10. has no local maximum or minimum, but 2 and 4 are critical numbers

(a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2. (b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2. (c) Sketch the graph of a function that has a local maximum at 2 and is not continuous at 2.

(a) Sketch the graph of a function on [-1, 2] that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on [-1, 2] that has a local maximum but no absolute maximum.

(a) Sketch the graph of a function on [-1, 2] that has an absolute maximum but no absolute minimum. (b) Sketch the graph of a function on [-1, 2] that is discontinuous but has both an absolute maximum and an absolute minimum.

(a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. (b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 15.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 16.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 17.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 18.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 19.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 20.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 21.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 22.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 23.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 24.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 25.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 26.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 27.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.) 28.

29–44 Find the critical numbers of the function. 29.

29–44 Find the critical numbers of the function. 30.

29–44 Find the critical numbers of the function. 31.

29–44 Find the critical numbers of the function. 32.

29–44 Find the critical numbers of the function. 33.

29–44 Find the critical numbers of the function. 34.

29–44 Find the critical numbers of the function. 35.

29–44 Find the critical numbers of the function. 36.

29–44 Find the critical numbers of the function. 37.

29–44 Find the critical numbers of the function. 38.

29–44 Find the critical numbers of the function. 39.

29–44 Find the critical numbers of the function. 40.

29–44 Find the critical numbers of the function. 41.

29–44 Find the critical numbers of the function. 42.

29–44 Find the critical numbers of the function. 43.

29–44 Find the critical numbers of the function. 44.

45–46 A formula for the derivative of a function is given. How many critical numbers does have? 45.

45–46 A formula for the derivative of a function is given. How many critical numbers does have? 46.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 47.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 48.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 49.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 50.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 51.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 52.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 53.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 54.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 55.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 56.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 57.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 58.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 59.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 60.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 61.

47–62 Find the absolute maximum and absolute minimum values of on the given interval. 62.

63. If and are positive numbers, find the maximum value of

64. Use a graph to estimate the critical numbers of correct to one decimal place.

65–68 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. 65.

65–68 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. 66.

65–68 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. 67.

65–68 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. 68.

Between 0? and 30?, the volume ( in cubic centimeters) of 1 kg of water at a temperature is given approximately by the formula Find the temperature at which water has its maximum density.

An object with weight is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is Where is a positive constant called the coefficient of friction and where 0 ? ? /2 . Show that is minimized when tan .

A model for the US average price of a pound of white sugar from 1993 to 2003 is given by the function Where t is measured in years since August of 1993. Estimate the times when sugar was cheapest and most expensive during the period 1993–2003.

On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. (a) Use a graphing calculator or computer to find the cubic polynomial that best models the velocity of the shuttle for the time interval . Then graph this polynomial. (b) Find a model for the acceleration of the shuttle and use it to estimate the maximum and minimum values of the acceleration during the first 125 seconds.

When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the airstream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity v of the airstream is related to the radius of the trachea by the equation where is a constant and is the normal radius of the trachea. The restriction on is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than is prevented (otherwise the person would suffocate). (a) Determine the value of in the interval [, ] at which v has an absolute maximum. How does this compare with experimental evidence? (b) What is the absolute maximum value of v on the interval? (c) Sketch the graph of on the interval [0, ]

Show that 5 is a critical number of the function But g does not have a local extreme value at 5.

Prove that the function has neither a local maximum nor a local minimum.

If f has a local minimum value at c , show that the function \(g(x)=-f(x)\) has a local maximum value at c. Text Transcription: g(x)=-f(x)

Prove Fermat’s Theorem for the case in which has a local minimum at .

A cubic function is a polynomial of degree 3; that is, it has the form , where a 0. (a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities. (b) How many local extreme values can a cubic function have?

Explain the difference between an absolute minimum and a local minimum.

Suppose is a continuous function defined on a closed interval [a, b]. (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for ? (b) What steps would you take to find those maximum and minimum values? Explain the difference between an absolute minimum and a local minimum.

3–4 For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.

3–4 For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.

5–6 Use the graph to state the absolute and local maximum and minimum values of the function.

5–6 Use the graph to state the absolute and local maximum and minimum values of the function.

7–10 Sketch the graph of a function that is continuous on [1, 5] and has the given properties. 7. Absolute minimum at 2, absolute maximum at 3, local minimum at 4

7–10 Sketch the graph of a function that is continuous on [1, 5] and has the given properties. 8. Absolute minimum at 1, absolute maximum at 5, local maximum at 2, local minimum at 4

7–10 Sketch the graph of a function that is continuous on [1, 5] and has the given properties. 9. Absolute maximum at 5, absolute minimum at 2, local maximum at 3, local minima at 2 and 4

7–10 Sketch the graph of a function that is continuous on [1, 5] and has the given properties. 10. has no local maximum or minimum, but 2 and 4 are critical numbers

(a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2. (b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2. (c) Sketch the graph of a function that has a local maximum at 2 and is not continuous at 2.

(a) Sketch the graph of a function on [1, 2] that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on [1, 2] that has a local maximum but no absolute maximum.

(a) Sketch the graph of a function on [1, 2] that has an absolute maximum but no absolute minimum. (b) Sketch the graph of a function on [1, 2] that is discontinuous but has both an absolute maximum and an absolute minimum.

(a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. (b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers.

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

15–28 Sketch the graph of by hand and use your sketch to find the absolute and local maximum and minimum values of . (Use the graphs and transformations of Sections 1.2 and 1.3.)

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29-44 Find the critical numbers of the function. \(h(t)=t^{3 / 4}-2 t^{1 / 4}\) Text Transcription: h(t) = t^3/4 - 2t^1/4

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

29–44 Find the critical numbers of the function.

45–46 A formula for the derivative of a function is given. How many critical numbers does have?

45–46 A formula for the derivative of a function is given. How many critical numbers does have?

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

47–62 Find the absolute maximum and absolute minimum values of on the given interval.

If a and b are positive numbers, find the maximum value of , .

Use a graph to estimate the critical numbers of correct to one decimal place.

65–68 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values.

65–68 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values.

65–68 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values.

65–68 (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values.

Between and , the volume ( in cubic centimeters) of 1 kg of water at a temperature is given approximately by the formula Find the temperature at which water has its maximum density.

An object with weight is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is where is a positive constant called the coefficient of friction and where . Show that is minimized when tan .

A model for the US average price of a pound of white sugar from 1993 to 2003 is given by the function where t is measured in years since August of 1993. Estimate the times when sugar was cheapest and most expensive during the period 1993–2003.

On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. (a) Use a graphing calculator or computer to find the cubic polynomial that best models the velocity of the shuttle for the time interval t [0, 125] . Then graph this polynomial. (b) Find a model for the acceleration of the shuttle and use it to estimate the maximum and minimum values of the acceleration during the first 125 seconds.

When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. For a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. The greater the velocity of the airstream, the greater the force on the foreign object. X rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. According to a mathematical model of coughing, the velocity v of the airstream is related to the radius of the trachea by the equation where is a constant and is the normal radius of the trachea. The restriction on is due to the fact that the tracheal wall stiffens under pressure and a contraction greater than is prevented (otherwise the person would suffocate). (a) Determine the value of in the interval at which has an absolute maximum. How does this compare with experimental evidence? (b) What is the absolute maximum value of on the interval? (c) Sketch the graph of on the interval

Show that 5 is a critical number of the function but g does not have a local extreme value at 5.

Prove that the function has neither a local maximum nor a local minimum.

If has a local minimum value at c, show that the function g(x) = - (x) has a local maximum value at c .

Prove Fermat’s Theorem for the case in which has a local minimum at .

A cubic function is a polynomial of degree 3; that is, it has the form , where . (a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities. (b) How many local extreme values can a cubic function have?

47–62 Find the absolute maximum and absolute minimum values of on the given interval. \(f(x)=x^3-6x^2+5,\quad\ [-3,\ 5]\)

47–62 Find the absolute maximum and absolute minimum values of on the given interval.