Use the derivative formulas and algebra to find the intervalswhere f(x)=(x + 50)/(x2 +

Indicate all critical points on the graph of f in Figure 4.12 and determine which correspond to local maxima of f, which to local minima, and which to neither.

Graph a function which has exactly one critical point, at x = 2, and exactly one inflection point, at x = 4.

Graph a function with exactly two critical points, one of which is a local minimum and the other is neither a local maximum nor a local minimum.

In Exercises 48, use derivatives to find the critical points and inflection points. f(x) = x3 9x2 + 24x + 5

In Exercises 48, use derivatives to find the critical points and inflection points. f(x) = x5 10x3 8

In Exercises 48, use derivatives to find the critical points and inflection points. f(x) = x5 + 15x4 + 25

In Exercises 48, use derivatives to find the critical points and inflection points. f(x)=5x 3 ln x

In Exercises 48, use derivatives to find the critical points and inflection points. f(x)=4xe3x

In Exercises 912, find all critical points and then use the firstderivative test to determine local maxima and minima. Check your answer by graphing. f(x)=3x4 4x3 + 6

In Exercises 912, find all critical points and then use the firstderivative test to determine local maxima and minima. Check your answer by graphing. f(x)=(x2 4)7

In Exercises 912, find all critical points and then use the firstderivative test to determine local maxima and minima. Check your answer by graphing. f(x)=(x3 8)4

In Exercises 912, find all critical points and then use the firstderivative test to determine local maxima and minima. Check your answer by graphing. f(x) = x x2 + 1

In Exercises 1314, find the critical points of the function and classify them as local maxima or local minima or neither. g(x) = xe3x

In Exercises 1314, find the critical points of the function and classify them as local maxima or local minima or neither. h(x) = x + 1/x

(a) Use a graph to estimate the x-values of any critical points and inflection points of f(x) = ex2 . (b) Use derivatives to find the x-values of any critical points and inflection points exactly.

In Exercises 1619, the function f is defined for all x. Use the graph of f to decide: (a) Over what intervals is f increasing? Decreasing? (b) Does f have local maxima or minima? If so, which, and where? 16

In Exercises 1619, the function f is defined for all x. Use the graph of f to decide: (a) Over what intervals is f increasing? Decreasing? (b) Does f have local maxima or minima? If so, which, and where? 17

In Exercises 1619, the function f is defined for all x. Use the graph of f to decide: (a) Over what intervals is f increasing? Decreasing? (b) Does f have local maxima or minima? If so, which, and where? 18

In Exercises 1619, the function f is defined for all x. Use the graph of f to decide: (a) Over what intervals is f increasing? Decreasing? (b) Does f have local maxima or minima? If so, which, and where? 19

(a) Show that if a is a positive constant, then x = 0 is the only critical point of f(x) = x + a x. (b) Use derivatives to show that f is increasing and its graph is concave down for all x > 0.

(a) If b is a positive constant and x > 0, find all critical points of f(x) = x b ln x. (b) Use the second-derivative test to determine whether the function has a local maximum or local minimum at each critical point.

(a) If a is a nonzero constant, find all critical points of f(x) = a x2 + x. (b) Use the second-derivative test to show that if a is positive then the graph has a local minimum, and if a is negative then the graph has a local maximum.

If U and V are positive constants, find all critical points of F(t) = Uet + V et .

Indicate on the graph of the derivative function f in Figure 4.13 the x-values that are critical points of the function f itself. At which critical points does f have local maxima, local minima, or neither?

Indicate on the graph of the derivative f in Figure 4.14 the x-values that are inflection points of the function f.

Indicate on the graph of the second derivative f in Figure 4.15 the x-values that are inflection points of the function f.

For Problems 2730, sketch a possible graph of y = f(x), using the given information about the derivatives y = f (x) and y = f(x). Assume that the function is defined and continuous for all real x. 27

For Problems 2730, sketch a possible graph of y = f(x), using the given information about the derivatives y = f (x) and y = f(x). Assume that the function is defined and continuous for all real x. 28

For Problems 2730, sketch a possible graph of y = f(x), using the given information about the derivatives y = f (x) and y = f(x). Assume that the function is defined and continuous for all real x. 29

For Problems 2730, sketch a possible graph of y = f(x), using the given information about the derivatives y = f (x) and y = f(x). Assume that the function is defined and continuous for all real x. 30

Suppose f has a continuous derivative whose values are given in the following table. (a) Estimate the x-coordinates of critical points of f for 0 x 10. (b) For each critical point, indicate if it is a local maximum of f, local minimum, or neither.

(a) The following table gives values of the differentiable function y = f(x). Estimate the x-values of critical points of f(x) on the interval 0 <x< 10. Classify each critical point as a local maximum, local minimum, or neither. (b) Now assume that the table gives values of the continuous function y = f (x) (instead of f(x)). Estimate and classify critical points of the function f(x).

If water is flowing at a constant rate (i.e., constant volume per unit time) into the vase in Figure 4.16, sketch a graph of the depth of the water against time. Mark on the graph the time at which the water reaches the corner of the vase.

If water is flowing at a constant rate (i.e., constant volume per unit time) into the Grecian urn in Figure 4.17, sketch a graph of the depth of the water against time. Mark on the graph the time at which the water reaches the widest point of the urn.

Find and classify the critical points of f(x) = x3(1x) 4 as local maxima and minima.

If m, n 2 are integers, find and classify the critical points of f(x) = xm(1 x) n.

The rabbit population on a small Pacific island is approximated by P = 2000 1 + e5.30.4t with t measured in years since 1774, when Captain James Cook left 10 rabbits on the island. (a) Graph P. Does the population level off? (b) Estimate when the rabbit population grew most rapidly. How large was the population at that time? (c) What natural causes could lead to the shape of the graph of P?

Find values of a and b so that the function f(x) = x2 + ax + b has a local minimum at the point (6, 5).

Find the value of a so that the function f(x) = xeax has a critical point at x = 3.

Find constants a and b in the function f(x) = axebx such that f( 1 3 )=1 and the function has a local maximum at x = 1 3 .

Graph f(x) = x + sin x, and determine where f is increasing most rapidly and least rapidly.

You might think the graph of f(x) = x2 + cos x should look like a parabola with some waves on it. Sketch the actual graph of f(x) using a calculator or computer. Explain what you see using f(x).

Problems 4344 show graphs of the three functions f, f , f. Identify which is which. 43

Problems 4344 show graphs of the three functions f, f , f. Identify which is which. 44

Problems 4546 show graphs of f, f , f. Each of these three functions is either odd or even. Decide which functions are odd and which are even. Use this information to identify which graph corresponds to f, which to f , and which to f. 45

Problems 4546 show graphs of f, f , f. Each of these three functions is either odd or even. Decide which functions are odd and which are even. Use this information to identify which graph corresponds to f, which to f , and which to f. 46

Use the derivative formulas and algebra to find the intervals where f(x)=(x + 50)/(x2 + 525) is increasing and the intervals where it is decreasing. It is possible, but difficult, to solve this problem by graphing f; describe the difficulty

Let f be a function with f(x) > 0 for all x. Set g = 1/f. (a) If f is increasing in an interval around x0, what about g? (b) If f has a local maximum at x1, what about g? (c) If f is concave down at x2, what about g?

In Problems 4950, the differentiable function f has x = 1 as its only zero and x = 2 as its only critical point. For the given functions, find all (a) Zeros (b) Critical points. y = f x2 3 

In Problems 4950, the differentiable function f has x = 1 as its only zero and x = 2 as its only critical point. For the given functions, find all (a) Zeros (b) Critical points. y = (f(x))2 + 3

In Problems 5152, the graph of f lies entirely above the xaxis and f (x) < 0 for all x. (a) Give the critical point(s) of the function, or explain how you know there are none. (b) Say where the function increases and where it decreases. y = (f(x))2

In Problems 5152, the graph of f lies entirely above the xaxis and f (x) < 0 for all x. (a) Give the critical point(s) of the function, or explain how you know there are none. (b) Say where the function increases and where it decreases. y = f x2

In Problems 5354, explain what is wrong with the statement. An increasing function has no inflection points.

In Problems 5354, explain what is wrong with the statement. For any function f, if f(0) = 0, there is an inflection point at x = 0.

In Problems 5557, give an example of: A function which has no critical points on the interval between 0 and 1.

In Problems 5557, give an example of: A function, f, which has a critical point at x = 1 but for which f (1) = 0.

In Problems 5557, give an example of: A function with local maxima and minima at an infinite number of points.

Are the statements in Problems 5866 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. A local minimum of f occurs at a critical point of f.

Are the statements in Problems 5866 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. If x = p is not a critical point of f, then x = p is not a local maximum of f.

Are the statements in Problems 5866 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. . A local maximum of f occurs at a point where f (x)=0.

Are the statements in Problems 5866 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. If x = p is not a local maximum of f, then x = p is not a critical point of f.

Are the statements in Problems 5866 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. If f (p)=0, then f(x) has a local minimum or local maximum at x = p.

Are the statements in Problems 5866 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. If f (x) is continuous and f(x) has no critical points, then f is everywhere increasing or everywhere decreasing.

Are the statements in Problems 5866 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. If f(x) is continuous and the graph of f has an inflection point at x = p, then f(p)=0.

Are the statements in Problems 5866 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. A critical point of f must be a local maximum or minimum of f.

Are the statements in Problems 5866 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. Every cubic polynomial has an inflection point.

In Problems 6772, give an example of a function f that makes the statement true, or say why such an example is impossible. Assume that f exists everywhere. f is concave up and f(x) is positive for all x.

In Problems 6772, give an example of a function f that makes the statement true, or say why such an example is impossible. Assume that f exists everywhere. f is concave down and f(x) is positive for all x.

In Problems 6772, give an example of a function f that makes the statement true, or say why such an example is impossible. Assume that f exists everywhere. f is concave down and f(x) is negative for all x.

In Problems 6772, give an example of a function f that makes the statement true, or say why such an example is impossible. Assume that f exists everywhere. f is concave up and f(x) is negative for all x.

In Problems 6772, give an example of a function f that makes the statement true, or say why such an example is impossible. Assume that f exists everywhere. f(x)f(x) < 0 for all x.

In Problems 6772, give an example of a function f that makes the statement true, or say why such an example is impossible. Assume that f exists everywhere. f(x)f (x)f(x)f(x) < 0 for all x. 1

Given that f (x) is continuous everywhere and changes from negative to positive at x = a, which of the following statements must be true? (a) a is a critical point of f(x) (b) f(a) is a local maximum (c) f(a) is a local minimum (d) f (a) is a local maximum (e) f (a) is a local minimum