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Popular >

extreme f(x)=|x^2-1|,-4<= x<= 4

Solution

extreme points

f (x) = x^{2} - 1, - 4 \leq x \leq 4

Solution

Maximum (- 4, 15), Minimum (- 1, 0), Maximum (0, 1), Minimum (1, 0), Maximum (4, 15)

Solution steps

Find the critical points:

x = - 4, x = - 1, x = 0, x = 1, x = 4

Find intervals:

Decreasing

: - 4 < x < - 1,

Increasing

: - 1 < x < 0,

Decreasing

: 0 < x < 1,

Increasing

: 1 < x < 4

Plug the extreme point

x = - 4

into

x^{2} - 1 \Rightarrow y = 15

Maximum (- 4, 15)

Plug the extreme point

x = - 1

into

x^{2} - 1 \Rightarrow y = 0

Minimum (- 1, 0)

Plug the extreme point

x = 0

into

x^{2} - 1 \Rightarrow y = 1

Maximum (0, 1)

Plug the extreme point

x = 1

into

x^{2} - 1 \Rightarrow y = 0

Minimum (1, 0)

Plug the extreme point

x = 4

into

x^{2} - 1 \Rightarrow y = 15

Maximum (4, 15)

Maximum (- 4, 15), Minimum (- 1, 0), Maximum (0, 1), Minimum (1, 0), Maximum (4, 15)

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