Solved: Highway Design A section of highway connecting two hillsides with grades of 6%

In Exercises 1–10, determine the open intervals on which the graph is concave upward or concave downward. \(y=x^{2}-x-2\) Equation Transcription: Text Transcription: y=x^2 - x - 2

In Exercises 1–10, determine the open intervals on which the graph is concave upward or concave downward. \(y=-x^{3}+3 x^{2}-2\) Equation Transcription: Text Transcription: y= -x^3 + 3x^2 - 2

In Exercises 1–10, determine the open intervals on which the graph is concave upward or concave downward. \(f(x)=\frac{24}{x^{2}+12}\) Equation Transcription: Text Transcription: f(x) = 24 / x^2 + 12

In Exercises 1–10, determine the open intervals on which the graph is concave upward or concave downward. \(f(x)=\frac{x^{2}-1}{2 x+1}\) Equation Transcription: Text Transcription: f(x) = x^2 - 1 / 2x + 1

In Exercises 1–10, determine the open intervals on which the graph is concave upward or concave downward. \(f(x)=\frac{x^{2}+1}{x^{2}-1}\) Equation Transcription: Text Transcription: f(x) = x^2 + 1 / x^2 - 1

In Exercises 1–10, determine the open intervals on which the graph is concave upward or concave downward. \(y=\frac{3 x^{5}+40 x^{3}+135 x}{270}\) Equation Transcription: Text Transcription: y = -3x^5 + 40x^3 + 135x / 270

In Exercises 1–10, determine the open intervals on which the graph is concave upward or concave downward. \(g(x)=3 x^{2}-x^{3}\) Equation Transcription: Text Transcription: g(x) = 3x^2 - x^3

In Exercises 110, determine the open intervals on which the graph is concave upward or concave downward.

In Exercises 1–10, determine the open intervals on which the graph is concave upward or concave downward. \(y=2 x-\tan x,\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) Equation Transcription: Text Transcription: y = 2x - tan x, (- pi/2 , pi/2)

In Exercises 1–10, determine the open intervals on which the graph is concave upward or concave downward. \(y=x+\frac{2}{\sin x},(-\pi, \pi)\) Equation Transcription: Text Transcription: y = x + 2 / sin x , (- pi, pi)

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.f
x x3  6x2  12x y

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.f
x 2x3  3x2  12x  5 f

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.f
x 4  8x  3 1 4x4  2x2 f

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.f
x 2x f
x 4  8x  3 1 4

In Exercises 11–28, find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=x(x-4)^{3}\) Equation Transcription: Text Transcription: f(x) = x(x - 4)^3

In Exercises 11–28, find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=x^{3}(x-2)\) Equation Transcription: Text Transcription: f(x) = x^3 (x - 2)

In Exercises 11–28, find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=x \sqrt{x+3}\) Equation Transcription: Text Transcription: f(x) = x sqrt x + 3

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.f
x xx  1 f

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.f
x x x2  1 f

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.f
x x  1 x f

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.f
x sin 0, 4 ,

In Exercises 11–28, find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=2 \csc \frac{3 x}{2},(0,2 \pi)\) Equation Transcription: Text Transcription: f(x) = 2 csc 3x/2, (0, 2 pi)

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.f
x sec
0, 4 x

In Exercises 11–28, find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=\sin x+\cos x,[0,2 \pi]\) Equation Transcription: Text Transcription: f(x) = sin x + cos x, [0, 2 pi]

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.f
x 2 sin x  sin 2x, 0, 2

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.f
x x  2 cos x, 0, 2

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.y x  ln x

In Exercises 1128, find the points of inflection and discuss the concavity of the graph of the function.y 1 2
ex  ex y x  ln x f

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x x 2  3x  8 4  4x3  2 y

In Exercises 29–54, find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=x^{2}+3 x-8\) Equation Transcription: Text Transcription: f(x) = x^2 + 3x - 8

In Exercises 29–54, find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=(x-5)^{2}\) Equation Transcription: Text Transcription: f(x) = (x - 5)^2

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x 
x  52 f
x

In Exercises 29–54, find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=x^{3}-3 x^{2}+3\) Equation Transcription: Text Transcription: f(x) = x^3 - 3x^2 + 3

In Exercises 29–54, find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=x^{3}-9 x^{2}+27 x\) Equation Transcription: Text Transcription: f(x) = x^3 - 9x^2 + 27x

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.g
x x2
6  x3 f

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.g
x 1 8
x  22
x  42 g
x

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x x 2  1 23  3 g

In Exercises 29–54, find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=\sqrt{x^{2}+1}\) Equation Transcription: Text Transcription: f(x) = sqrt x^2 + 1

In Exercises 29–54, find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=x+\frac{4}{x}\) Equation Transcription: Text Transcription: f(x) =x + 4/x

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x x x  1 f

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x cos x  x, 0, 4

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x 2 sin x  cos 2x, 0, 2 f

In Exercises 29–54, find all relative extrema. Use the Second Derivative Test where applicable. \(y=\frac{1}{2} x^{2}-\ln x\) Equation Transcription: Text Transcription: y = 1/2 x^2 - ln x

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.y x ln x 1

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.y x ln x

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.y x2 ln x 4 y

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x ex  ex 2 y

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.g
x 1 2e
x322 f
x

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x x2 ex g

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x xex f

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x 8x
4x

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.y x2 log3 x

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x arcsec x  x y

In Exercises 2954, find all relative extrema. Use the Second Derivative Test where applicable.f
x arcsin x  2x f

In Exercises 5558, use a computer algebra system to analyze the function over the given interval. (a) Find the first and second derivatives of the function. (b) Find any relative extrema and points of inflection. (c) Graph and on the same set of coordinate axes and state the relationship between the behavior of and the signs of andf
.

In Exercises 5558, use a computer algebra system to analyze the function over the given interval. (a) Find the first and second derivatives of the function. (b) Find any relative extrema and points of inflection. (c) Graph and on the same set of coordinate axes and state the relationship between the behavior of and the signs of andf
. f
x x 6, 6 26  x2 ,

In Exercises 5558, use a computer algebra system to analyze the function over the given interval. (a) Find the first and second derivatives of the function. (b) Find any relative extrema and points of inflection. (c) Graph and on the same set of coordinate axes and state the relationship between the behavior of and the signs of andf
. f
x sin x  0,  1 3 sin 3x  1 5 sin 5x, f

In Exercises 5558, use a computer algebra system to analyze the function over the given interval. (a) Find the first and second derivatives of the function. (b) Find any relative extrema and points of inflection. (c) Graph and on the same set of coordinate axes and state the relationship between the behavior of and the signs of andf
. f
x 2x sin x, 0, 2

Consider a function such that is increasing. Sketch graphs of for (a) and ((b) (b)

Consider a function such that is decreasing. Sketch graphs of for (a) and (b)Consider a function such that is decreasing. Sketch graphs of for (a) and (b)

Sketch the graph of a function that does have a point of inflection at even though

S represents weekly sales of a product. What can be said of and for each of the following? (a) The rate of change of sales is increasing. (b) Sales are increasing at a slower rate. (c) The rate of change of sales is constant. (d) Sales are steady. (e) Sales are declining, but at a slower rate. (f) Sales have bottomed out and have started to rise. S S S

In Exercises 63–66, the graph of \(f\) is shown. Graph \(f\), \(f^{\prime}\), \(f^{\prime \prime}\) and on the same set of coordinate axes. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Equation Transcription: Text Transcription: f f^prime f^prime ^prime

In Exercises 63–66, the graph of \(f\) is shown. Graph \(f\), \(f^{\prime}\), \(f^{\prime \prime}\) and on the same set of coordinate axes. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Equation Transcription: Text Transcription: f f^prime f^prime ^prime

In Exercises 63–66, the graph of \(f\) is shown. Graph \(f\), \(f^{\prime}\), \(f^{\prime \prime}\) and on the same set of coordinate axes. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Equation Transcription: Text Transcription: f f^prime f^prime ^prime

In Exercises 63–66, the graph of \(f\) is shown. Graph \(f\), \(f^{\prime}\), \(f^{\prime \prime}\) and on the same set of coordinate axes. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Equation Transcription: Text Transcription: f f^prime f^prime ^prime

Think About It In Exercises 67–70, sketch the graph of a function having the given characteristics. f(2) = f(4) = 0 f(3) is defined. \(f^{\prime}(x)\) < 0 if x < 3 \(f^{\prime}(3)\) does not exist. \(f^{\prime}(x)\) > 0 if x > 3 \(f^{\prime \prime}(x)<0, x \neq 3\) Equation Transcription: Text Transcription: f ^prime(x) f ^prime(3) f ^prime ^prime(x) < 0, x neq 3

Think About It In Exercises 67–70, sketch the graph of a function having the given characteristics. f(0) = f(2) = 0 \(f^{\prime}(x)\) > 0 if x < 1 \(f^{\prime}(1)\) = 0 \(f^{\prime}(x)\) < 0 if x > 1 \(f^{\prime \prime}(x)\) < 0 Equation Transcription: Text Transcription: f ^prime(x) f ^prime(1) f ^prime ^prime(x)

Think About It In Exercises 67–70, sketch the graph of a function having the given characteristics. f(2) = f(4) = 0 \(f^{\prime}(x)\) > 0 if x < 3 \(f^{\prime}(3)\) does not exist. \(f^{\prime}(x)\) < 0 if x > 3 \(f^{\prime \prime}(x)>0, x \neq 3\) Equation Transcription: Text Transcription: f ^prime(x) f ^prime(3) f ^prime ^prime(x) > 0, x neq 3

Think About It In Exercises 67–70, sketch the graph of a function having the given characteristics. f(0) = f(2) = 0 \(f^{\prime}(x)\) < 0 if x < 1 \(f^{\prime}(1)\) = 0 \(f^{\prime}(x)\) > 0 if x > 1 \(f^{\prime \prime}(x)\) > 0 Equation Transcription: Text Transcription: f ^prime(x) f ^prime(1) f ^prime ^prime(x)

Conjecture Consider the function (a) Use a graphing utility to graph for 2, 3, and 4. Use the graphs to make a conjecture about the relationship between and any inflection points of the graph of (b) Verify your conjecture in part (a).

(a) Graph and identify the inflection point. (b) Does exist at the inflection point? Explain.

In Exercises 73 and 74, find \(a\), \(b\), \(c\) and \(d\) such that the cubic \(f(x)=a x^{3}+b x^{2}+c x+d\) satisfies the given conditions. Relative maximum: (3, 3) Relative minimum: (5, 1) Inflection point: (4, 2) Equation Transcription: Text Transcription: a b c d f(x) = ax^3 + bx^2 + cx + d

In Exercises 73 and 74, find \(a\), \(b\), \(c\) and \(d\) such that the cubic \(f(x)=a x^{3}+b x^{2}+c x+d\) satisfies the given conditions. Relative maximum: (2, 4) Relative minimum: (4, 2) Inflection point: (3, 3) Equation Transcription: Text Transcription: a b c d f(x) = ax^3 + bx^2 + cx + d

Aircraft Glide Path A small aircraft starts its descent from an altitude of 1 mile, 4 miles west of the runway (see figure). (a) Find the cubic on the interval that describes a smooth glide path for the landing. (b) The function in part (a) models the glide path of the plane. When would the plane be descending at the most rapid rate?

Highway Design A section of highway connecting two hillsides with grades of 6% and 4% is to be built between two points that are separated by a horizontal distance of 2000 feet (see figure). At the point where the two hillsides come together, there is a 50-foot difference in elevation. (a) Design a section of highway connecting the hillsides modeled by the function At the points and the slope of the model must match the grade of the hillside. (b) Use a graphing utility to graph the model. (c) Use a graphing utility to graph the derivative of the model. (d) Determine the grade at the steepest part of the transitional section of the highway.

Beam Deflection The deflection of a beam of length is where is the distance from one end of the beam. Find the value of that yields the maximum deflection

Specific Gravity A model for the specific gravity of water is where is the water temperature in degrees Celsius. (a) Use a computer algebra system to find the coordinates of the maximum value of the function. (b) Sketch a graph of the function over the specified domain. (Use a setting in which (c) Estimate the specific gravity of water when T 20 .

Average Cost A manufacturer has determined that the total cost of operating a factory is where is the number of units produced. At what level of production will the average cost per unit be minimized? (The average cost per unit is Cx.

Modeling Data The average typing speed (words per minute) of a typing student after weeks of lessons is shown in the table. A model for the data is (a) Use a graphing utility to plot the data and graph the model. (b) Use the second derivative to determine the concavity of Compare the result with the graph in part (a). (c) What is the sign of the first derivative for By combining this information with the concavity of the model, what inferences can be made about the typing speed as increases?

Linear and Quadratic Approximations In Exercises 81–84, use a graphing utility to graph the function. Then graph the linear and quadratic approximations \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) in the same viewing window. Compare the values of \(f\), \(P_{1}\) and \(P_{2}\) and their first derivatives at x = a. How do the approximations change as you move farther away from x = a? FUNCTION Value of a f(x) = 2(sin x + cos x) \(a=\frac{\pi}{4}\) Equation Transcription: Text Transcription: P_1 (x) = f(a) + f ^prime (a)(x-a) P_2 (x) = f(a) + f ^prime (a)(x-a) + 1/2 f^prime ^prime (a)(x-a)^2 f P_1 P_2 a = pi/4

Linear and Quadratic Approximations In Exercises 81–84, use a graphing utility to graph the function. Then graph the linear and quadratic approximations \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) in the same viewing window. Compare the values of \(f\), \(P_{1}\) and \(P_{2}\) and their first derivatives at x = a. How do the approximations change as you move farther away from x = a? FUNCTION Value of a f(x) = 2(sin x + cos x) a = 0 Equation Transcription: Text Transcription: P_1 (x) = f(a) + f ^prime (a)(x-a) P_2 (x) = f(a) + f ^prime (a)(x-a) + 1/2 f^prime ^prime (a)(x-a)^2 f P_1 P_2

Linear and Quadratic Approximations In Exercises 81–84, use a graphing utility to graph the function. Then graph the linear and quadratic approximations \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) in the same viewing window. Compare the values of \(f\), \(P_{1}\) and \(P_{2}\) and their first derivatives at x = a. How do the approximations change as you move farther away from x = a? FUNCTION Value of a f(x) = arctan x a = -1 Equation Transcription: Text Transcription: P_1 (x) = f(a) + f ^prime (a)(x-a) P_2 (x) = f(a) + f ^prime (a)(x-a) + 1/2 f^prime ^prime (a)(x-a)^2 f P_1 P_2

Linear and Quadratic Approximations In Exercises 81–84, use a graphing utility to graph the function. Then graph the linear and quadratic approximations \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) in the same viewing window. Compare the values of \(f\), \(P_{1}\) and \(P_{2}\) and their first derivatives at x = a. How do the approximations change as you move farther away from x = a? FUNCTION Value of a \(f(x)=\frac{\sqrt{x}}{x-1}\) a = 2 Equation Transcription: Text Transcription: P_1 (x) = f(a) + f ^prime (a)(x-a) P_2 (x) = f(a) + f ^prime (a)(x-a) + 1/2 f^prime ^prime (a)(x-a)^2 f P_1 P_2 f(x) = sqrt x / x - 1

Use a graphing utility to graph Show that the graph is concave downward to the right ofx 1. y

Show that the point of inflection of lies midway between the relative extrema of f

Prove that every cubic function with three distinct real zeros has a point of inflection whose -coordinate is the average of the three zeros.

Show that the cubic polynomial has exactly one point of inflection where and Use this formula to find the point of inflection of p
x x3  3x2  2. y

True or False? In Exercises 8994, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The graph of every cubic polynomial has precisely one point of inflection

True or False? In Exercises 8994, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. . The graph of is concave downward for and concave upward for and thus it has a point of inflection at x 0.

True or False? In Exercises 8994, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The maximum value of sin cos is 5.

True or False? In Exercises 8994, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The maximum slope of the graph of is b

True or False? In Exercises 8994, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. f then is concave upward at x =c.

True or False? In Exercises 8994, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If then the graph of must have a point of inflection at x =2.

In Exercises 95 and 96, let and represent differentiable functions such that andg
 0. Show that if and are concave upward on the interval then is also concave upward on a, b. f

In Exercises 95 and 96, let and represent differentiable functions such that andg
 0. Prove that if and are positive, increasing, and concave upward on the interval then is also concave upward on
a, b. f