Free Printable Maxima and Minima

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Maxima and Minima

Total questions: 25

Worksheet time: 3hrs 32mins

Name

Class

Date

Does this function have a minimum, maximum, both or neither?

Minimum

Maximum

Both

Neither

If a function has a second derivative that is negative, what does that tell you?

The function is increasing

The function is decreasing

The function is concave up

The function is concave down

If a function has a second derivative that is positive, what does that tell you?

The function is increasing

The function is decreasing

The function is concave up

The function is concave down

What is a point of inflection?

When a function goes from increasing to decreasing

When a function goes from concave up to concave down

The slope of a function is described by its ____________.

first derivative

second derivative

third derivative

expression

Does this graph have a maximum or minimum value?

maximum

minimum

neither

both

Does
y = -4x²
have a maximum or minimum value?

maximum

minimum

both

neither

Find the x-coordinate of the inflection point of f(x)=6x²-x³.

-2

None

Find the critical points of f(x) = 2x⁴- 4x² + 1

x= 0

x = -1, 1

x = -1, 0, 1

no critical points

10.

Identify the relative maximum:

(0, 3)

(4, 1)

(3, 0)

(1, 4)

11.

A critical point of the function f occurs at an interior point c

(a) only when f ′(c) = 0

(b) only when f ′(c) fails to exist.

12.

If f ′(c) = 0 , then

(a) f has a local maximum or local minimum at c.

(b) f has an absolute maximum or minimum at c.

13.

Which of the following statements is true about the function f (x) = x⁵ ?

(a) f ′(c) = 0 .

(b) f has a local minimum at x = 0.

14.

On the interval [−2, 2] the function f (x) = x⁴

(a) has an absolute maximum but no local maxima.

(b) has an absolute maximum at an interior point of the interval.

15.

If f '(3) = 0 and f"(3) < 0, then which of the following must be true?

There is a local max at x=3

There is a local min at x = 3

There is an inflection point at x = 3

There is an x-intercept at x = 3

16.

The concavity of a function is described by its _______________.

first derivative

second derivative

third derivative

expression

17.

What is the maximum value of f(x) = x³ - 3x² - 1 on the interval [-3, 2]?

-1

18.

Find the point(s) of inflection for f(x) = 2(x)^1/5 + 3

None

x = 0

x = 0, 2

x = -2, 0, 2

19.

What is the maximum or minimum point for the curve

y = x² − 4 ?

A minimum at ( 0, -4)

A minimum at ( 0, 4)

A maximum at ( 0, -4)

A maximum at ( 0, 4)

20.

What is the maximum or minimum point for the curve y = -3x² − 12x + 5 ?

A minimum at ( - 2, 17 )

A minimum at ( 2, -31 )

a maximum at ( - 2, 17 )

A maximum at ( 2, -31 )

21.

Find the critical points of the given function: f(x) = x³-3x²

( 0,2 )

(0,0) and (2,-4)

( 0,-4 ) (2,0 )

( 0,2 ) ( 3,0 )

22.

Find the critical points of the following function:

f (x ) = 4x/(x² + 1)

(-1, - 2) and (1, 2)

(1, 2) , (-1, 2) and (0,0)

(2, 1) , (0, 1) and (-2, - 1)

(-1, 2) and (1, -2)

23.

For what values of x does the graph of g have a point of inflection f(x) = x⁴ + 4x³ - 18x² ?

x = 4 and x = -1

x = -4 and x = 1

x = -3 and x = 1

x = 3 and x = -1

24.

Determine the inflection points of the curve

𝑦 = 𝑥 + 2 𝑥 − 5 ?

( 2, 3 )

( 0, 2 )

(-1, 1 )

( no inflection point )

25.

Find the inflection points of the function

𝑓 ( 𝑥 ) = (1/2 )x⁴ - 3x² + 4

( 1, - 2 ) and ( -1 , 2 )

( 1, 11/2 ) and ( -1 , 11/2 )

( 1, - 5/2 ) and ( -1 , 5/2 )

( 1, 3/2 ) and ( -1 , 3/2 )

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Maxima and Minima

Total questions: 25

Worksheet time: 3hrs 32mins

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