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Applied Calculus For Business, Economics, and Finance

Warren B. Gordon, Walter O. Wang, April Allen Materowski

Chapter 3

Applications of the Derivative

Educators


Problem 1

Sketch the graph of a function which is increasing to the left of $x=1$ and decreasing to its right.

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Problem 2

Sketch the graph of a function which is decreasing to the left of $x=1,$ increasing to its right and passes through (0,-3)

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Problem 3

Sketch the graph of a continuous function increasing on $-1<x<2$ and decreasing on $2<x<4$. Indicate the point $M$ on your graph which is a relative maximum.

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Numerade Educator

Problem 4

Sketch the graph of a continuous function decreasing on $-5<x<3$ and increasing on $3<x<7 .$ Indicate the point $m$ on your graph which is a relative minimum.

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Numerade Educator

Problem 5

Sketch the graph of a continuous function defined on $-5 \leq x \leq 3$, that has relative maxima at $(x=-1)$ and $(x=3)$ and a relative minimum at $(x=0)$

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Problem 6

Sketch the graph of a continuous function that has a relative maximum at $(1,1 / 2)$ a minimum at (-2,-5) and a maximum at (-3,1)

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Problem 7

Which of the points in Figure 12 are: (a) maxima? (b) minima? (c) relative maxima? (d) relative minima?

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Problem 8

(a) Draw the graph of a function which has two maxima and two relative maxima. (b) What must be true about the $y$ -values at the maxima?

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Problem 9

Draw the graph of a function such that the minimum is also a relative minimum.

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Problem 10

Draw the graph of a function such that the maximum is also a relative maximum.

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Problem 11

Locate all critical points.
$$f(x)=x^{2}-2 x+3$$

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Problem 12

Locate all critical points.
$$f(x)=4 x^{5}$$

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Problem 13

Locate all critical points.
$$g(x)=a x^{2}+b x+c$$

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Problem 14

Locate all critical points.
$$s(t)=2 t^{3}-9 t^{2}-60 t+5$$

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Problem 15

Locate all critical points.
$$h(x)=4 x^{3}-13 x^{2}+12 x+9$$

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Numerade Educator

Problem 16

Locate all critical points.
$$f(x)=2 x^{4}+2 x^{3}-x^{2}-7$$

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Problem 17

Locate all critical points.
$$h(x)=\left(12-x^{2}\right)^{1 / 2}$$

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Problem 18

Locate all critical points.
$$r(x)=4 x^{3 / 4}+2$$

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Problem 19

Locate all critical points.
$$v(t)=t^{6}-3 t^{2}+5$$

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Problem 20

Locate all critical points.
$$s(t)=(t-1)^{4}(t+3)^{3}$$

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Problem 21

Locate all critical points.
$$f(x)=\frac{x+3}{x-3}$$

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Problem 22

Locate all critical points.
$$w(x)=\frac{3 x}{4 x^{2}+9}$$

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Problem 23

Locate all critical points.
$$h(x)=\frac{1}{x^{2}-x-2}$$

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Problem 24

Locate all critical points.
$$f(x)=\left(x^{2}-9\right)^{2 / 3}$$

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Problem 25

Locate all critical points.
$$w(x)=\frac{x}{\sqrt{x-4}}$$

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Problem 26

Locate all critical points.
$$g(x)=\frac{x^{2}}{x^{2}-9}$$

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Problem 27

Locate all critical points.
$$f(x)=4-x^{2 / 3}$$

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Problem 28

Determine the extrema on the given interval.
$$f(x)=3 x+9 \text { on: }(a)[-1,3] ;(b)(-1,3)$$

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Numerade Educator

Problem 29

Determine the extrema on the given interval.
$$f(x)=x^{2}-2 x+3 \text { on: }(a)[0,2] ;(b)[2,3] ;(c)(2,3]$$

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Numerade Educator

Problem 30

Determine the extrema on the given interval.
$$f(x)=2 x^{3}+3 x^{2}-12 x-6 \text { on: }(a)[-3,2] ;(b)[-5,3]$$

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Numerade Educator

Problem 31

Determine the extrema on the given interval.
$$f(x)=4 x^{3 / 4}+2 \text { on: }(a)[0,16] ;(b)(0,16) ;(c)[0,16)$$

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Numerade Educator

Problem 32

Determine the extrema on the given interval.
$$f(x)=x /\left(x^{2}+1\right) \text { on: }(a)[0,2] ;(b)[-2,2] ;(c)(0,2) ;(d)(-2,2)$$

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Numerade Educator

Problem 33

Determine the extrema on the given interval.
$$f(x)=x^{4}-6 x^{3}+12 x^{2}+2 \text { on: }(a)[1,2] ;(b)[-1,2]$$

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Numerade Educator

Problem 34

Determine the extrema on the given interval.
$$f(x)=x \sqrt{2-x} \text { on }[0,3 / 2]$$

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Problem 35

Let $f(x)$ be defined on the closed interval [0,1] by the rule:
$$f(x)=\left\{\begin{array}{cl}
2 x^{2} & \text { if } 0<x<1 \\
1 & \text { if } x=0 \text { or } x=1
\end{array}\right.$$
(a) Does fhave extrema on [0,1]$?$ (b) Is this a violation of Theorem $1 ?$

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Problem 36

Let $f(x)=x^{-1 / 2}$ on the interval (0,1] . (a) Does $f$ satisfy the conditions of Theorem $1 ?$ (b) Does it have a maximum value? (c) Does it have a minimum value?

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Problem 37

Let $f(x)=x^{-2 / 3}$ on the interval $[-1,8] .$ Does $f$ have extrema on this interval?

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Problem 38

Show, by means of a sketch, that if a continuous function with a single critical point that is a relative extremum, then this critical point is also an extremum.

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Problem 39

Determine the extrema of the function defined by the equation $f(x)=\frac{x}{x^{2}+1}$ on $(-\infty, 0] .$ Justify your conclusions.

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Problem 40

(a) Determine the extrema of the function defined by the equation $f(x)=\frac{x^{2}}{x^{3}+1}$ on $[0, \infty) .$ Justify your conclusions.
(b) Does this function have extrema on $(-\infty, 0] ?$

Wendi Z.
Numerade Educator