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Popular >

extreme f(x)=x^4-2x^2+8,-4<= x<= 4

Solution

extreme points

f (x) = x^{4} - 2 x^{2} + 8, - 4 \leq x \leq 4

Solution

Maximum (- 4, 232), Minimum (- 1, 7), Maximum (0, 8), Minimum (1, 7), Maximum (4, 232)

Solution steps

Find the critical points:

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

Domain of

x^{4} - 2 x^{2} + 8, - 4 \leq x \leq 4 : - 4 \leq x \leq 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^{4} - 2 x^{2} + 8 \Rightarrow y = 232

Maximum (- 4, 232)

Plug the extreme point

x = - 1

into

x^{4} - 2 x^{2} + 8 \Rightarrow y = 7

Minimum (- 1, 7)

Plug the extreme point

x = 0

into

x^{4} - 2 x^{2} + 8 \Rightarrow y = 8

Maximum (0, 8)

Plug the extreme point

x = 1

into

x^{4} - 2 x^{2} + 8 \Rightarrow y = 7

Minimum (1, 7)

Plug the extreme point

x = 4

into

x^{4} - 2 x^{2} + 8 \Rightarrow y = 232

Maximum (4, 232)

Maximum (- 4, 232), Minimum (- 1, 7), Maximum (0, 8), Minimum (1, 7), Maximum (4, 232)

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