?Coffee is being poured into the mug shown in the figure at a constant rate (measured in

Use the given graph of \(f\) to find the following. (a) The open intervals on which \(f\) is increasing. (b) The open intervals on which \(f\) is decreasing. (c) The open intervals on which \(f\) is concave upward. (d) The open intervals on which \(f\) is concave downward. (e) The coordinates of the points of inflection. Equation Transcription: Text Transcription: f f f f f

Use the given graph of \(f\) to find the following. (a) The open intervals on which \(f\) is increasing. (b) The open intervals on which \(f\) is decreasing. (c) The open intervals on which \(f\) is concave upward. (d) The open intervals on which \(f\) is concave downward. (e) The coordinates of the points of inflection. Equation Transcription: Text Transcription: f f f f f

Suppose you are given a formula for a function \(f\). (a) How do you determine where \(f\) is increasing or decreasing? (b) How do you determine where the graph of \(f\) is concave upward or concave downward? (c) How do you locate inflection points? Equation Transcription: Text Transcription: f f f

(a) State the First Derivative Test. (b) State the Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?

The graph of the derivative \(f^{\prime}\) of a function \(f\) is shown. (a) On what intervals is \(f\) increasing? Decreasing? (b) At what values of \(x\) does \(f\) have a local maximum? Local minimum? Equation Transcription: Text Transcription: f' f f x f

The graph of the derivative \(f^{\prime}\) of a function \(f\) is shown. (a) On what intervals is \(f\) increasing? Decreasing? (b) At what values of \(x\) does \(f\) have a local maximum? Local minimum? Equation Transcription: Text Transcription: f' f f x f

In each part state the \(x\)-coordinates of the inflection points of \(f\). Give reasons for your answers. (a) The curve is the graph of \(f\). (b) The curve is the graph of \(f^{\prime}\). (c) The curve is the graph of \(f^{\prime \prime}\). Equation Transcription: ’ Text Transcription: x-coordinates f f f’ f"

The graph of the first derivative \(f^{\prime}\) of a function \(f\) is shown. (a) On what intervals is \(f\) increasing? Explain. (b) At what values of \(x\) does \(f\) have a local maximum or minimum? Explain. (c) On what intervals is \(f\) concave upward or concave downward? Explain. (d) What are the \(x\)-coordinates of the inflection points of \(f\) ? Why? Equation Transcription: Text Transcription: f' f f x f f x-coordinates f

Find the intervals on which f is increasing or decreasing, and find the local maximum and minimum values of f. \(f(x)=2 x^{3}-15 x^{2}+24 x-5\)

Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=x^{3}-6 x^{2}-135 x\) Equation Transcription: Text Transcription: f f f(x)=x^3-6x^2-135x

Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=6 x^{4}-16 x^{3}+1\) Equation Transcription: Text Transcription: f f f(x)=6x^4-16x^3+1

Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=x^{2 / 3}(x-3)\) Equation Transcription: Text Transcription: f f f(x)=x^2/3(x-3)

Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=\frac{x^{2}-24}{x-5}\) Equation Transcription: Text Transcription: f f f(x)=x^2 - 24/x - 5

Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=x+\frac{4}{x^{2}}\) Equation Transcription: Text Transcription: f f f(x)=x+4/x^2

Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=\sin x+\cos x, \quad 0 \leqslant x \leqslant 2 \pi\) Equation Transcription: ? ? Text Transcription: f f f(x)=sin x+cos x, 0 leqslant x leqslant 2pi

Find the intervals on which \(f\) is increasing or decreasing, and find the local maximum and minimum values of \(f\). \(f(x)=x^{4} e^{-x}\) Equation Transcription: Text Transcription: f f f(x)=x^4 e^-x

Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=x^{3}-3 x^{2}-9 x+4\) Equation Transcription: Text Transcription: f f f(x)=x^3-3x^2-9x+4

Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=2 x^{3}-9 x^{2}+12 x-3\) Equation Transcription: Text Transcription: f f f(x)=2x^3-9x^2+12x-3

Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=\sin ^{2} x-\cos 2 x, \quad 0 \leqslant x \leqslant \pi\) Equation Transcription: ? ? Text Transcription: f f f(x)=sin^2 x-cos 2x, 0 leqslant x leqslant pi

Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=\ln (2+\sin x), \quad 0 \leqslant x \leqslant 2 \pi\) Equation Transcription: ? ? Text Transcription: f f f(x)=ln(2+sin x), 0 leqslant x leqslant 2pi

Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=\ln \left(x^{2}+5\right)\) Equation Transcription: Text Transcription: f f f(x)=ln(x^2+5)

Find the intervals on which \(f\) is concave upward or concave downward, and find the inflection points of \(f\). \(f(x)=\frac{e^{x}}{e^{x}+2}\) Equation Transcription: Text Transcription: f f f(x)=e^x/e^x + 2

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f\). (c) Find the intervals of concavity and the inflection points. \(f(x)=x^{4}-2 x^{2}+3\) Equation Transcription: Text Transcription: f f f(x)=x^4-2x^2+3

(a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find the local maximum and minimum values of \(f\). (c) Find the intervals of concavity and the inflection points. \(f(x)=\frac{x}{x^{2}+1}\) Equation Transcription: Text Transcription: f F f(x)=x/x^2 + 1

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.

(a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. \(f(x)=x^{2} \ln x\)

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.

(a) Find the intervals on which is increasing or decreasing. (b) Find the local maximum and minimum values of . (c) Find the intervals of concavity and the inflection points.

Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? \(f(x)=1+3 x^{2}-2 x^{3}\)

Find the local maximum and minimum values of using both the First and Second Derivative Tests. Which method do you prefer?

Suppose the derivative of a function f is \(f^{\prime}(x)=(x-4)^{2}(x+3)^{7}(x-5)^{8}\) On what interval(s) is f increasing?

(a) Find the critical numbers of . (b) What does the Second Derivative Test tell you about the behavior of at these critical numbers? (c) What does the First Derivative Test tell you?

Suppose is continuous on . (a) If and , what can you say about ? (b) If and , what can you say about ?

Sketch the graph of a function that satisfies all of the given conditions. (a) and for all (b) and for all

Sketch the graph of a function that satisfies all of the given conditions. (a) and for all (b) and for all

Sketch the graph of a function that satisfies all of the given conditions. Vertical asymptote if , if , if if

Sketch the graph of a function that satisfies all of the given conditions. , if or , if or , if if or

Sketch the graph of a function that satisfies all of the given conditions. for all , vertical asymptote , if or , if

Sketch the graph of a function that satisfies all of the given conditions. when , when , when or , for

Sketch the graph of a function that satisfies all of the given conditions.

Sketch the graph of a function that satisfies all of the given conditions. if if , if , has inflection point ,

The graph of a function is shown. At which point(s) are the following true? (a) and are both positive. (b) and are both negative. (c) is negative but is positive.

The graph of the derivative ’ of a continuous function is shown. (a) On what intervals is increasing? Decreasing? (b) At what values of does have a local maximum? Local minimum? (c) On what intervals is concave upward? Concave downward? (d) State the x-coordinate(s) of the point(s) of inflection. (e) Assuming that , sketch a graph of .

The graph of the derivative ’ of a continuous function is shown. (a) On what intervals is increasing? Decreasing? (b) At what values of does have a local maximum? Local minimum? (c) On what intervals is concave upward? Concave downward? (d) State the x-coordinate(s) of the point(s) of inflection. (e) Assuming that , sketch a graph of .

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)-(c) to sketch the graph. You may want to check your work with a graphing calculator or computer. \(f(x)=36 x+3 x^{2}-2 x^{3}\)

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) - (c) to sketch the graph. You may want to check your work with a graphing calculator or computer. \(f(x)=\frac{1}{2} x^4-4 x^2+3\)

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)-(c) to sketch the graph. You may want to check your work with a graphing calculator or computer. \(g(t)=3 t^{4}-8 t^{3}+12\)

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) - (c) to sketch the graph. You may want to check your work with a graphing calculator or computer. \(f(x)=\left(x^2-4\right)^3\)

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer. C(x) = x1/3(x + 4)

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer. f(x) = ln(x2 + 9)

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.

(a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a) – (c) to sketch the graph. You may want to check your work with a graphing calculator or computer.

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f. \(f(x)=1+\frac{1}{x}-\frac{1}{x^{2}}\)

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f. \(f(x)=x-\frac{1}{6} x^{2}-\frac{2}{3} \ln x\)

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a) – (d) to sketch the graph of f.

Use the methods of this section to sketch several members of the given family of curves. What do the members have in common? How do they differ from each other?

Use the methods of this section to sketch several members of the given family of curves. What do the members have in common? How do they differ from each other?

(a) Use a graph of f to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of x at which f increases most rapidly. Then find the exact value.

(a) Use a graph of f to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of x at which f increases most rapidly. Then find the exact value. \(f(x)=x^{2} e^{-x}\)

(a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of to give better estimates.

(a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of to give better estimates.

Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph .

Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph .

A graph of a population of yeast cells in a new laboratory culture as a function of time is shown. (a) Describe how the rate of population increase varies. (b) When is this rate highest? (c) On what intervals is the population function concave upward or downward? (d) Estimate the coordinates of the inflection point.

In an episode of The Simpsons television show, Homer reads from a newspaper and announces “Here’s good news! According to this eye-catching article, SAT scores are declining at a slower rate.” Interpret Homer’s statement in terms of a function and its first and second derivatives.

The president announces that the national deficit is increasing, but at a decreasing rate. Interpret this statement in terms of a function and its first and second derivatives.

Let f(t) be the temperature at time t where you live and suppose that at time t = 3 you feel uncomfortably hot. How do you feel about the given data in each case? (a) (b) (c) (d)

Let K(t) be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, K(8) - K(7) or K(3) - K(2)? Is the graph of K concave upward or concave downward? Why?

Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity. What is the significance of the inflection point?

A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S(t) ? Atpe-kt is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A = 0.01, p = 4, k = 0.07, and t is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. Then graph the drug response curve.

Normal Density Functions The family of bell-shaped curves occurs in probability and statistics, where it is called the normal density function. The constant is called the mean and the positive constant is called the standard deviation. For simplicity, let’s scale the function so as to remove the factor and let’s analyze the special case where = 0. So we study the function (a) Find the asymptote, maximum value, and inflection points of f. (b) What role does play in the shape of the curve? (c) Illustrate by graphing four members of this family on the same screen.

Find a cubic function \(f(x)=a x^{3}+b x^{2}+c x+d\) that has a local maximum value of 3 at x = -2 and a local minimum value of 0 at x = 1.

For what values of the numbers a and b does the function have the maximum value f(2) = 1?

Show that the curve y = (1 + x) / (1 + x2) has three points of inflection and they all lie on one straight line.

Show that the curves \(y=e^{-x}\) and \(y=-e^{-x}\) touch the curve \(y=e^{-x} \sin x\) at its inflection points.

Show that the inflection points of the curve y = x sin x lie on the curve \(y^{2}\left(x^{2}+4\right)=4 x^{2}\).

Assume that all of the functions are twice differentiable and the second derivatives are never 0. (a) If f and g are concave upward on an interval I, show that f + g is concave upward on I. (b) If f is positive and concave upward on I, show that the function g(x) = [f(x)]2 is concave upward on I.

Assume that all of the functions are twice differentiable and the second derivatives are never 0. (a) If f and g are positive, increasing, concave upward functions on an interval I, show that the product function fg is concave upward on I. (b) Show that part (a) remains true if f and g are both decreasing. (c) Suppose f is increasing and g is decreasing. Show, by giving three examples, that fg may be concave upward, concave downward, or linear. Why doesn’t the argument in parts (a) and (b) work in this case?

Assume that all of the functions are twice differentiable and the second derivatives are never 0. Suppose f and g are both concave upward on (-?, ?). Under what condition on f will the composite function h(x) ? f(g(x)) be concave upward?

Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three x-intercepts x1, x2, and x3, show that the x-coordinate of the inflection point is (x1 + x2 + x3)/3

For what values of c does the polynomial P(x) + x4 + cx3 + x2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as c decreases?

Prove that if (c, f(c)) is a point of inflection of the graph of f and \(f^{\prime \prime}\) exists in an open interval that contains c, then \(f^{\prime \prime}(c)=0\) [Hint. Apply the First Derivative Test and Fermat's Theorem to the function \(g=f^{\prime}\).]

Show that if f(x) = x4, then , but (0, 0) is not an inflection point of the graph of f.

Show that the function g(x) = x | x | has an inflection point at (0, 0) but does not exist.

Suppose that is continuous and = = 0, but . Does f have a local maximum or minimum at c? Does f have a point of inflection at c?

Suppose f is differentiable on an interval I and \(f^{\prime}(x)>0\) for all numbers x in I except for a single number c. Prove that f is increasing on the entire interval I.

For what values of c is the function \(f(x)=c x+\frac{1}{x^{2}+3}\) increasing on \((-\infty, \infty)\) ?

The three cases in the First Derivative Test cover the situations commonly encountered but do not exhaust all possibilities. Consider the functions f, g, and h whose values at 0 are all 0 and, for x 0, (a) Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0. (b) Show that f has neither a local maximum nor a local minimum at 0, g has a local minimum, and h has a local maximum.