Answer: In Exercises 2758, find the critical points and the intervals on which the

For which value of m is the following statement correct? If f (2) = 3 and f (4) = 9, and f is differentiable, then f has a tangent line of slope m.

Assume f is differentiable. Which of the following statements does not follow from the MVT? (a) If f has a secant line of slope 0, then f has a tangent line of slope 0. (b) If f (5)< f(9), then f (c) > 0 for some c (5, 9). (c) If f has a tangent line of slope 0, then f has a secant line of slope 0. (d) If f (x) > 0 for all x, then every secant line has positive slope. 3

Can a function with the real numbers as its domain that takes on only negative values have a positive derivative? If so, sketch an example.

For f with derivative as in Figure 13: (a) Is f (c) a local minimum or maximum? (b) Is f a decreasing function? c x y FIGURE 13 Graph of derivative f .

In Exercises 18, find a point c satisfying the conclusion of the MVT for the given function and interval.y = x3, [4, 5]

In Exercises 18, find a point c satisfying the conclusion of the MVT for the given function and interval.y = x ln x, [1, 2]

In Exercises 18, find a point c satisfying the conclusion of the MVT for the given function and interval.y = e2x , [0, 3]

In Exercises 18, find a point c satisfying the conclusion of the MVT for the given function and interval.y = ex x, [1, 1]

In Exercises 912, find a point c satisfying the conclusion of the MVT for the given function and interval. Then draw the graph of the function, the secant line between the endpoints of the graph and the tangent line at (c, f (c)), to see that the secant and tangent lines are, in fact, parallel. 9. y = x2, [0, 1]

In Exercises 912, find a point c satisfying the conclusion of the MVT for the given function and interval. Then draw the graph of the function, the secant line between the endpoints of the graph and the tangent line at (c, f (c)), to see that the secant and tangent lines are, in fact, parallel.y = x2/3, [0, 8]

In Exercises 912, find a point c satisfying the conclusion of the MVT for the given function and interval. Then draw the graph of the function, the secant line between the endpoints of the graph and the tangent line at (c, f (c)), to see that the secant and tangent lines are, in fact, parallel.y = ex , [0, 1]

In Exercises 912, find a point c satisfying the conclusion of the MVT for the given function and interval. Then draw the graph of the function, the secant line between the endpoints of the graph and the tangent line at (c, f (c)), to see that the secant and tangent lines are, in fact, parallel.y = x, [0, 3]

Let f (x) = x5 + x2. The secant line between x = 0 and x = 1 has slope 2 (check this), so by the MVT, f (c) = 2 for some c (0, 1). Plot f and the secant line on the same axes. Then plot y = 2x + b for different values of b until the line becomes tangent to the graph of f . Zoom in on the point of tangency to estimate the x-coordinate c of the point of tangency.

Plot the derivative of f (x) = 3x5 5x3. Describe its sign changes and use this to determine the local extreme values of f . Then graph f to confirm your conclusions.

Determine the intervals on which f (x) is positive and negative, assuming that Figure 14 is the graph of f .

Determine the intervals on which f is increasing or decreasing, assuming that Figure 14 is the graph of f .

State whether f (2) and f (4) are local minima or local maxima, assuming that Figure 14 is the graph of f .

Figure 15 shows the graph of the derivative f of a function f . Find the critical points of f and determine whether they are local minima, local maxima, or neither. 2 1 0.5 2 3 y = f (x) 6 2 y

In Exercises 1922, sketch the graph of a function f whose derivative f has the given description. 19. f (x) > 0 for x > 3 and f (x) < 0 for x < 3 20.

In Exercises 1922, sketch the graph of a function f whose derivative f has the given description.f (x) > 0 for x < 1 and f (x) < 0 for x > 1 21

In Exercises 1922, sketch the graph of a function f whose derivative f has the given description.f (x) is negative on (1, 3) and positive everywhere else.

In Exercises 1922, sketch the graph of a function f whose derivative f has the given description.f (x) makes the sign transitions +, , +, .

In Exercises 2326, find all critical points of f and use the First Derivative Test to determine whether they are local minima or maxima. 23. f (x) = 4 + 6x x2

In Exercises 2326, find all critical points of f and use the First Derivative Test to determine whether they are local minima or maxima.f (x) = x3 12x 4

In Exercises 2326, find all critical points of f and use the First Derivative Test to determine whether they are local minima or maxima.f (x) = x2 x + 1

In Exercises 2326, find all critical points of f and use the First Derivative Test to determine whether they are local minima or maxima.f (x) = x3 + x3

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither). 27. y = x2 + 7x 17

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = 5x2 + 6x 4

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x3 12x2

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x(x 2)3

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = 3x4 + 8x3 6x2 24x

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x2 + (10 x)2

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = 1 3 x3 + 3 2 x2 + 2x + 4

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x4 + x3

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x5 + x3 + 1

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x5 + x3 + x

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x4 4x3/2 (x > 0)

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x5/2 x2 (x > 0)

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x + x1 (x > 0)

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x2 4x1 (x > 0)

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = 1 x2 + 1

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = 2x + 1 x2 + 1

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x3 x2 + 1

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x3 x2 3

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = + sin + cos

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = sin + 3 cos

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = sin2 + sin

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = 2 cos , [0, 2]

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x + ex

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = ex x (x > 0)

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = ex cos x,  2 , 2

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x2ex

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = tan1 x 1 2 x

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = (x2 2x)ex

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x ln x (x > 0)

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = ln x x (x > 0)

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x1/3

In Exercises 2758, find the critical points and the intervals on which the function is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local min or max (or neither).y = x2/3 x2

Find the minimum value of f (x) = xx for x > 0.

Show that f (x) = x2 + bx + c is decreasing on , b 2 and increasing on b 2 , .

Show that f (x) = x3 2x2 + 2x is an increasing function. Hint: Find the minimum value of f . 6

Find conditions on a and b that ensure f (x) = x3 + ax + b is increasing on (,).

Let h(x) = x(x2 1) x2 + 1 and suppose that f (x) = h(x). Plot h and use the plot to describe the local extrema and the increasing/ decreasing behavior of f . Sketch a plausible graph for f itself.

Sam made two statements that Deborah found dubious. (a) The average velocity for my trip was 70 mph; at no point in time did my speedometer read 70 mph. (b) Apoliceman clocked me going 70 mph, but my speedometer never read 65 mph. In each case, which theorem did Deborah apply to prove Sams statement false: the Intermediate Value Theorem or the Mean Value Theorem? Explain.

Determine where f (x) = (1,000 x)2 + x2 is decreasing. Use this to decide which is larger: 8002 + 2002 or 6002 + 4002.

Show that f (x) = 1 |x| satisfies the conclusion of the MVT on [a, b] if both a and b are positive or negative, but not if a < 0 and b > 0.

Which values of c satisfy the conclusion of the MVT on the interval [a, b] if f is a linear function?

Show that if f is any quadratic polynomial, then the midpoint c = a + b 2 satisfies the conclusion of the MVT on [a, b] for any a and b.

Suppose that f (0) = 2 and f (x) 3 for x > 0. Apply the MVT to the interval [0, 4] to prove that f (4) 14. Prove more generally that f (x) 2 + 3x for all x > 0.

Show that if f (2) = 2 and f (x) 5 for x > 2, then f (4) 8.

Show that if f (2) = 5 and f (x) 10 for x > 2, then f (x) 10x 15 for all x > 2.

Show that a cubic function f (x) = x3 + ax2 + bx + c is increasing on (,) if b>a2/3.

Prove that if f (0) = g(0) and f (x) g (x) for x 0, then f (x) g(x) for all x 0. Hint: Show that the function given by y = f (x) g(x) is nonincreasing. 7

Use Exercise 73 to prove that x tan x for 0 x < 2 .

Use Exercise 73 to prove that x tan x for 0 x < 2 .(b) sin x x 1 6 x3 (c) cos x 1 1 2 x2 + 1 24 x4 Can you guess the next inequality in the series?

Let f (x) = ex . Use the method of Exercise 75 to prove the following inequalities for x 0: (a) ex 1 x (b) ex 1 x + 1 2 x2 (c) ex 1 x + 1 2 x2 1 6 x3 Can you guess the next inequality in the series?

Assume that f exists and f (x) = 0 for all x. Prove that f (x) = mx + b, where m = f (0) and b = f (0). 78.

Define f (x) = x3 sin 1 x for x = 0 and f (0) = 0. (a) Show that f is continuous at x = 0 and that x = 0 is a critical point of f . (b) Examine the graphs of f and f . Can the First Derivative Test be applied? (c) Show that f (0) is neither a local min nor a local max. 7

Suppose that f (x) satisfies the following equation (an example of a differential equation): f (x) = f (x) 1 (a) Show that f (x)2 + f (x)2 = f (0)2 + f (0)2 for all x. Hint: Show that the function on the left has zero derivative. (b) Verify that sin x and cos x satisfy Eq. (1), and deduce that sin2 x + cos2 x = 1. 80.

Suppose that functions f and g satisfy Eq. (1) and have the same initial valuesthat is, f (0) = g(0) and f (0) = g (0). Prove that f (x) = g(x) for all x. Hint: Apply Exercise 79(a) to f g. 8

Use Exercise 80 to prove f (x) = sin x is the unique solution of Eq. (1) such that f (0) = 0 and f (0) = 1; and g(x) = cos x is the unique solution such that g(0) = 1 and g (0) = 0. This result can be used to develop all the properties of the trigonometric functions analyticallythat is, without reference to triangles. 4