Practice Optimization in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The process of using derivatives to systematically find maximum or minimum values of a function over a domain.

Find where the function hits its peaks (maxima) and valleys (minima) by finding where the slope is zero.

Showing a random 20 of 50 problems.

Example 1

easy

Find the absolute minimum of

f(x) = (x-3)^2 + 1

Example 2

hard

Find the dimensions of the right circular cone of maximum volume that can be inscribed in a sphere of radius

R

Example 3

medium

Find two non-negative numbers whose sum is

20

and whose product is maximum.

Example 4

hard

Find the absolute minimum value of

f(x) = x^2 + \frac{16}{x}

for

x > 0

Example 5

medium

Find the dimensions of a rectangle with perimeter 20 that maximizes area.

Example 6

easy

Where does

f(x) = x^2

attain its minimum?

Example 7

medium

Find the absolute maximum and minimum of

f(x) = x^3 - 6x^2 + 9x + 1

[0, 4]

Example 8

medium

Find the absolute extrema of

f(x) = \sin x + \cos x

[0, 2\pi]

Example 9

hard

A right triangle with the right angle at the origin has legs along the positive axes. The hypotenuse passes through

(3, 4)

. Find the minimum area of the triangle.

Example 10

easy

Find the critical points of

f(x) = x^3 - 3x

Example 11

easy

Find the absolute max of

f(x) = 4 - x^2

[-1, 1]

Example 12

medium

A poster has area

180

^2

. If margins are

3

cm on top and bottom and

2

cm on each side, what poster dimensions maximize the printed area?

Example 13

medium

A page must contain

50

^2

of printed text, with margins of

1

in on top/bottom and

2

in on each side. What page dimensions minimize total page area?

Example 14

easy

What does the second derivative test tell you if

f'(c) = 0

and

f''(c) > 0

Example 15

challenge

Snell's law: a swimmer at

(0, 4)

on land swims to point

(x, 0)

on a straight shoreline, then swims in the water to

(10, -3)

. Land speed

5

m/s, water speed

3

m/s. Set up the time function

T(x)

and write the equation that

x

must satisfy (don't solve numerically).

Example 16

medium

A box with a square base and open top has volume 32. Minimize surface area.

Example 17

medium

Classify the critical point of

f(x) = -x^2 + 6x

using the second derivative test.

Example 18

challenge

A 12 cm wire is cut into two pieces, one bent into a square and one into a circle. Where to cut to minimize total area? (Set up the critical-point equation.)

Example 19

easy

x=0

a max, min, or neither for

f(x) = x^3

Example 20

challenge

Maximize the area of a rectangle inscribed under

y = 4 - x^2

(above the

x

-axis), with base on the axis.

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Related Concepts

Derivative