Practice Monotonicity in Math

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

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Quick Recap

A function or sequence that consistently moves in one direction only—always increasing or always decreasing throughout its domain.

Your age is monotonically increasing—it only goes up, never back down. A timer counting down is monotonically decreasing.

Showing a random 20 of 50 problems.

Example 1

challenge

f: \mathbb{R} \to \mathbb{R}

is monotonically increasing AND satisfies

f(x+y) = f(x) + f(y)

for all

x, y

, what form must

f

take?

Example 2

easy

f(x) = -x^3

monotonic on

\mathbb{R}

? If so, increasing or decreasing?

Example 3

easy

Is the temperature reading throughout a day (rising in morning, falling at night) monotonic?

Example 4

medium

Is the function

g(x) = x + \sin x

monotonically increasing on

\mathbb{R}

Example 5

challenge

f(x)=\frac{x}{x+1}

increasing or decreasing for

x>-1

? Justify.

Example 6

easy

f(x) = 7

(a constant) strictly increasing?

Example 7

easy

f(x)=-2x+5

increasing or decreasing?

Example 8

easy

Is the sequence

1, 4, 9, 16, 25, \dots

monotonically increasing?

Example 9

medium

f

is increasing and

g

is increasing, must

f \circ g

be increasing?

Example 10

medium

Is the sequence

a_n=\frac{1}{n}

for

n=1,2,3,\dots

increasing or decreasing?

Example 11

easy

f(x) = \ln x

x > 0

— increasing or decreasing?

Example 12

medium

f(x)=x^3

monotonic on all reals?

Example 13

medium

For

f(x) = -x + 5

, is it increasing or decreasing? Verify with two test values.

Example 14

hard

Find values of

a

for which

f(x) = x^3 + ax + 1

is monotonically increasing on all of

\mathbb{R}

Example 15

medium

For

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

, on what interval is

f

increasing?

Example 16

hard

f(x) = x + \frac{1}{x}

monotonic on

(0, \infty)

? If not, where is it increasing / decreasing?

Example 17

challenge

Prove that if a sequence

(a_n)

is monotonically increasing and bounded above, it converges.

Example 18

medium

f

is increasing and

f(2)=5

f(7)=9

, what can you say about

f(4)

Example 19

easy

f(x)=\frac{1}{x}

for

x>0

increasing or decreasing?

Example 20

hard

f

is decreasing and

g

is decreasing, must

f \circ g

be increasing or decreasing?

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Related Concepts

Function Inverse Function