In Exercises 53 58, use a graphing utility to (a) graph the function on the given

In Exercises 1–4, explain why Rolle’s Theorem does not apply to the function even though there exist \(a\) and \(b\) such that f(a) = f(b). f(x) = 1 - |x - 1| Equation Transcription: Text Transcription: a b

In Exercises 1–4, explain why Rolle’s Theorem does not apply to the function even though there exist \(a\) and \(b\) such that f(a) = f(b). \(f(x)=\cot \frac{x}{2}\) Equation Transcription: Text Transcription: a b f(x) = cot x/2

In Exercises 1–4, explain why Rolle’s Theorem does not apply to the function even though there exist \(a\) and \(b\) such that f(a) = f(b). \(f(x)=\left|\frac{1}{x}\right|\), [-1, 1] Equation Transcription: Text Transcription: a b f(x) = |1/x|

In Exercises 1–4, explain why Rolle’s Theorem does not apply to the function even though there exist \(a\) and \(b\) such that f(a) = f(b). \(f(x)=\sqrt{\left(2-x^{2 / 3}\right)^{3}}\), [-1, 1] Equation Transcription: Text Transcription: a b f(x) = sqrt (2 - x^2/3)^3

In Exercises 58, find the two -intercepts of the function and show that at some point between the two x-intercepts.

In Exercises 58, find the two -intercepts of the function and show that at some point between the two x-intercepts. f
x
x
x 3 2

In Exercises 58, find the two -intercepts of the function and show that at some point between the two x-intercepts. f
x
x x  4 f

In Exercises 58, find the two -intercepts of the function and show that at some point between the two x-intercepts. f
x
3x x  1 f

In Exercises 9 and 10, the graph of \(f\) is shown. Apply Rolle’s Theorem and find all values of \(c\) such that \(f^{\prime}(c)=0\) at some point between the labeled intercepts. Equation Transcription: Text Transcription: f c f’(c) = 0

In Exercises 9 and 10, the graph of \(f\) is shown. Apply Rolle’s Theorem and find all values of \(c\) such that \(f^{\prime}(c)=0\) at some point between the labeled intercepts. Equation Transcription: Text Transcription: f c f’(c) = 0

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x
x 0, 2 2

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x
x 1, 4 f

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x

x 1
x 2
x 3, 1, 3 f
x
x

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0.f
x

x 3
x  1 1, 3 2 ,

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x
x 8, 8 , 2

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x
3 0, 6 x x 0, 2 2

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x
1, 3 x2 2x 3 x  2 , f
x

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x
1, 1 x2 1 x , f

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0.f
x

x 0, 2 f

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x
x 2 ln x, 1, 3 2

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x
sin x, 0, 2 f

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x
cos x, 0, 2 f

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. 6 f

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0.
x
cos 2x, 

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x
tan x, 0,  f

In Exercises 1126, determine whether Rolles Theorem can be applied to on the closed interval If Rolles Theorem can be applied, find all values of in the open interval such that f
c 0. f
x
sec x,  4 , 

In Exercises 2732, use a graphing utility to graph the function on the closed interval Determine whether Rolles Theorem can be applied to f on the interval and, if so, find all values of c in the open interval such that f
c 0. f
x
1, 1 , x

In Exercises 2732, use a graphing utility to graph the function on the closed interval Determine whether Rolles Theorem can be applied to f on the interval and, if so, find all values of c in the open interval such that f
c 0. f
x
x x 0, 1 13

In Exercises 2732, use a graphing utility to graph the function on the closed interval Determine whether Rolles Theorem can be applied to f on the interval and, if so, find all values of c in the open interval such that f
c 0. f
x
4x tan x,  1 4, 1 4 f

In Exercises 2732, use a graphing utility to graph the function on the closed interval Determine whether Rolles Theorem can be applied to f on the interval and, if so, find all values of c in the open interval such that f
c 0. f
x
1, 0 x 2 sin x 6 , 1 1, 1 , x

In Exercises 2732, use a graphing utility to graph the function on the closed interval Determine whether Rolles Theorem can be applied to f on the interval and, if so, find all values of c in the open interval such that f
c 0. f
x
2  arcsin
x 1, 1 2 1, f
x

In Exercises 2732, use a graphing utility to graph the function on the closed interval Determine whether Rolles Theorem can be applied to f on the interval and, if so, find all values of c in the open interval such that f
c 0. f
x
2 
x 0, 4 2 4x
2x4 , f
x

Vertical Motion The height of a ball seconds after it is thrown upward from a height of 32 feet and with an initial velocity of 48 feet per second is (a) Verify that (b) According to Rolles Theorem, what must be the velocity at some time in the interval Find that time.

Reorder Costs The ordering and transportation cost of components used in a manufacturing process is approximated by where is measured in thousands of dollars and is the order size in hundreds. (a) Verify that (b) According to Rolles Theorem, the rate of change of cost must be 0 for some order size in the interval Find that order size.

In Exercises 35 and 36, copy the graph and sketch the secant line to the graph through the points (a, f(a)) and (b, f(b)). Then sketch any tangent lines to the graph for each value of \(c\) guaranteed by the Mean Value Theorem. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Equation Transcription: Text Transcription: c

In Exercises 35 and 36, copy the graph and sketch the secant line to the graph through the points (a, f(a)) and (b, f(b)). Then sketch any tangent lines to the graph for each value of \(c\) guaranteed by the Mean Value Theorem. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. Equation Transcription: Text Transcription: c

In Exercises 37–40, explain why the Mean Value Theorem does not apply to the function \(f\) on the interval [0, 6]. Equation Transcription: Text Transcription: f

In Exercises 37–40, explain why the Mean Value Theorem does not apply to the function \(f\) on the interval [0, 6]. Equation Transcription: Text Transcription: f

In Exercises 37–40, explain why the Mean Value Theorem does not apply to the function \(f\) on the interval [0, 6]. \(f(x)=\frac{1}{x-3}\) Equation Transcription: Text Transcription: f f(x) = 1/x - 3

In Exercises 37–40, explain why the Mean Value Theorem does not apply to the function \(f\) on the interval [0, 6]. f(x) = |x - 3| Equation Transcription: Text Transcription: f

Consider the graph of the function \(f(x)=x^{2}+1\). (a) Find the equation of the secant line joining the points (-1, 2) and (2, 5). (b) Use the Mean Value Theorem to determine a point \(c\) in the interval (-1, 2) such that the tangent line at \(c\) is parallel to the secant line. (c) Find the equation of the tangent line through \(c\). (d) Use a graphing utility to graph \(f\), the secant line, and the tangent line. Equation Transcription: Text Transcription: f(x) = x^2 + 1 c f

Consider the graph of the function \(f(x)=-x^{2}-x+6\). (a) Find the equation of the secant line joining the points (-2, 4) and (2, 0). (b) Use the Mean Value Theorem to determine a point \(c\) in the interval (-2, 2) such that the tangent line at \(c\) is parallel to the secant line. (c) Find the equation of the tangent line through \(c\). (d) Use a graphing utility to graph \(f\), the secant line, and the tangent line. Equation Transcription: Text Transcription: f(x) = -x^2 - x + 6 c f

In Exercises 4352, determine whether the Mean Value Theorem can be applied to on the closed interval If the Mean Value Theorem can be applied, find all values of in the open interval such that f
c f b fa b a a,f
x
x 2, 1 2,

In Exercises 4352, determine whether the Mean Value Theorem can be applied to on the closed interval If the Mean Value Theorem can be applied, find all values of in the open interval such that f
c f b fa b a a, f
x
x
x 1, 1 2 x

In Exercises 4352, determine whether the Mean Value Theorem can be applied to on the closed interval If the Mean Value Theorem can be applied, find all values of in the open interval such that f
c f b fa b a a, f
x
x 0, 1 , 2

In Exercises 4352, determine whether the Mean Value Theorem can be applied to on the closed interval If the Mean Value Theorem can be applied, find all values of in the open interval such that f
c f b fa b a a,f
x
, 2 x  1 x f

In Exercises 4352, determine whether the Mean Value Theorem can be applied to on the closed interval If the Mean Value Theorem can be applied, find all values of in the open interval such that f
c f b fa b a a, f
x
2 x, 7, 2 , 1

In Exercises 4352, determine whether the Mean Value Theorem can be applied to on the closed interval If the Mean Value Theorem can be applied, find all values of in the open interval such that f
c f b fa b a a, f
x
x 0, 1 3

In Exercises 4352, determine whether the Mean Value Theorem can be applied to on the closed interval If the Mean Value Theorem can be applied, find all values of in the open interval such that f
c f b fa b a a, f
x
sin x, 0,  f

In Exercises 4352, determine whether the Mean Value Theorem can be applied to on the closed interval If the Mean Value Theorem can be applied, find all values of in the open interval such that f
c f b fa b a a, f
x
2 sin x  sin 2x, 0,  f

In Exercises 4352, determine whether the Mean Value Theorem can be applied to on the closed interval If the Mean Value Theorem can be applied, find all values of in the open interval such that f
c f b fa b a a, f
x
x log 1, 2 2

In Exercises 4352, determine whether the Mean Value Theorem can be applied to on the closed interval If the Mean Value Theorem can be applied, find all values of in the open interval such that f
c f b fa b a a,f
x
arctan
1 x, 0, 1 f
x

In Exercises 53 58, use a graphing utility to (a) graph the function on the given interval, (b) find and graph the secant line through points on the graph of at the endpoints of the given interval, and (c) find and graph any tangent lines to the graph of that are parallel to the secant line. f
x
, 2 x

In Exercises 53 58, use a graphing utility to (a) graph the function on the given interval, (b) find and graph the secant line through points on the graph of at the endpoints of the given interval, and (c) find and graph any tangent lines to the graph of that are parallel to the secant line. f
x
x 2 sin x, ,  1

In Exercises 53 58, use a graphing utility to (a) graph the function on the given interval, (b) find and graph the secant line through points on the graph of at the endpoints of the given interval, and (c) find and graph any tangent lines to the graph of that are parallel to the secant line. f
x
x, 1, 9

In Exercises 53 58, use a graphing utility to (a) graph the function on the given interval, (b) find and graph the secant line through points on the graph of at the endpoints of the given interval, and (c) find and graph any tangent lines to the graph of that are parallel to the secant line. f
x
x 0, 5 4  4x3  8x2  5, f

In Exercises 53 58, use a graphing utility to (a) graph the function on the given interval, (b) find and graph the secant line through points on the graph of at the endpoints of the given interval, and (c) find and graph any tangent lines to the graph of that are parallel to the secant line. f
x
2e 0, 2 ,

In Exercises 53 58, use a graphing utility to (a) graph the function on the given interval, (b) find and graph the secant line through points on the graph of at the endpoints of the given interval, and (c) find and graph any tangent lines to the graph of that are parallel to the secant line. f
x
ln  sec x

Let be continuous on and differentiable on If there exists in such that does it follow that Explain.

Let be continuous on the closed interval and differentiable on the open interval Also, suppose that and that is a real number in the interval such that Find an interval for the function over which Rolles Theorem can be applied, and find the corresponding critical number of ( is a constant).

The function is differentiable on and satisfies However, its derivative is never zero on Does this contradict Rolles Theorem? Explain.

Can you find a function such that f
2
6, and for all Why or why not? f
x < 1 x? f f
2
2,

Speed A plane begins its takeoff at 2:00 P.M. on a 2500-mile flight. The plane arrives at its destination at 7:30 P.M. Explain why there are at least two times during the flight when the speed of the plane is 400 miles per hour

Temperature When an object is removed from a furnace and placed in an environment with a constant temperature of its core temperature is Five hours later the core temperature is Explain why there must exist a time in the interval when the temperature is decreasing at a rate of per hour.

Velocity Two bicyclists begin a race at 8:00 A.M. They both finish the race 2 hours and 15 minutes later. Prove that at some time during the race, the bicyclists are traveling at the same velocity.

Acceleration At 9:13 A.M., a sports car is traveling 35 miles per hour. Two minutes later, the car is traveling 85 miles per hour. Prove that at some time during this two-minute interval, the cars acceleration is exactly 1500 miles per hour squared.

Graphical Reasoning The figure shows two parts of the graph of a continuous differentiable function on The derivative is also continuous. To print an enlarged copy of the graph, go to the website www.mathgraphs.com. (a) Explain why must have at least one zero in (b) Explain why must also have at least one zero in the interval What are these zeros called? (c) Make a possible sketch of the function with one zero of on the interval (d) Make a possible sketch of the function with two zeros of on the interval (e) Were the conditions of continuity of and necessary to do parts (a) through (d)? Explain.

Consider the function (a) Use a graphing utility to graph and (b) Is a continuous function? Is a continuous function? (c) Does Rolles Theorem apply on the interval Does it apply on the interval Explain. (d) Evaluate, if possible, and lim x3 lim f
x. x

In Exercises 69 and 70, sketch the graph of an arbitrary function \(f\) that satisfies the given condition but does not satisfy the conditions of the Mean Value Theorem on the interval [-5, 5]. \(f\) is continuous on [-5, 5]. Equation Transcription: Text Transcription: f

In Exercises 69 and 70, sketch the graph of an arbitrary function \(f\) that satisfies the given condition but does not satisfy the conditions of the Mean Value Theorem on the interval [-5, 5]. \(f\) is not continuous on [-5, 5]. Equation Transcription: Text Transcription: f

In Exercises 71 and 72, use the Intermediate Value Theorem and Rolles Theorem to prove that the equation has exactly one real solution.

In Exercises 71 and 72, use the Intermediate Value Theorem and Rolles Theorem to prove that the equation has exactly one real solution. 2x 2 cos x
0 5

Determine the values of and such that the function satisfies the hypotheses of the Mean Value Theorem on the interva 0, 3. f
x
 1, ax  b, x2  4x  c, x
0 0 < x 1 1 < x 3 0

Determine the values of and such that the function satisfies the hypotheses of the Mean Value Theorem on the interva -1, 2. f
x
 a, 2, bx2  c, dx  4, x
1 1 < x 0 0 < x 1 1 < x 2 1,

Differential Equations In Exercises 7578, find a function that has the derivative and whose graph passes through the given point. Explain your reasoning. f
x
0,
2, 5 f
x

Differential Equations In Exercises 7578, find a function that has the derivative and whose graph passes through the given point. Explain your reasoning. f
x
4,
0, 1 f

Differential Equations In Exercises 7578, find a function that has the derivative and whose graph passes through the given point. Explain your reasoning. f
x
2x,
1, 0 f
x

Differential Equations In Exercises 7578, find a function that has the derivative and whose graph passes through the given point. Explain your reasoning.f
x
2x  3,
1, 0 f

True or False? In Exercises 7982, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The Mean Value Theorem can be applied to on the interval -1, 1.

True or False? In Exercises 7982, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of a function has three -intercepts, then it must have at least two points at which its tangent line is horizontal.

True or False? In Exercises 7982, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of a polynomial function has three -intercepts, then it must have at least two points at which its tangent line is horizonta

True or False? In Exercises 7982, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If for all in the domain of then is a constant function.

The Mean Value Theorem can be applied to on the interval -1, 1.Prove that if and is any positive integer, then the polynomial function cannot have two real roots.

Prove that if for all in an interval then is constant on
a, b.

Let Prove that for any interval the value guaranteed by the Mean Value Theorem is the midpoint of the interval.

(a) Let and Then and Show that there is at least one value in the interval where the tangent line to at is parallel to the tangent line to at Identify (b) Let and be differentiable functions on where and Show that there is at least one value in the interval where the tangent line to at is parallel to the tangent line to at c, g
c. c

Prove that if is differentiable on and for all real numbers, then has at most one fixed point. A fixed point of a function is a real number such that f
c
c.

Use the result of Exercise 87 to show that has at most one fixed point.

Prove that \(|\cos a-\cos b| \ \leq\ |a-b|\) for all \(a\) and \(b\). Equation Transcription: Text Transcription: |cos a - cos b| leq |a - b| a b

Prove that \(|\sin a-\sin b| \ \leq\ |a-b|\) for all \(a\) and \(b\). Equation Transcription: Text Transcription: |sin a - sin b| leq |a - b| a b

Let Use the Mean Value Theorem to show that b a < b a 2 a .