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Popular >

extreme (1-x^3)/(x^2-4)

Solution

extreme points

\frac{1 - x ^{3}}{x ^{2} - 4}

Solution

Minimum (- 3.54460 \dots, 5.31691 \dots), Maximum (0, - \frac{1}{4}), Minimum (0.16705 \dots, - 0.25058 \dots), Maximum (3.37755 \dots, - 5.06632 \dots)

Solution steps

Find the critical points:

x = - 3.54460 \dots, x = 0, x \approx 0.16705 \dots, x \approx 3.37755 \dots

Domain of

\frac{1 - x ^{3}}{x ^{2} - 4} : x < - 2 or - 2 < x < 2 or x > 2

Find intervals:

Decreasing

: - \infty < x < - 3.54460 \dots,

Increasing

: - 3.54460 \dots < x < - 2,

Increasing

: - 2 < x < 0,

Decreasing

: 0 < x < 0.16705 \dots,

Increasing

: 0.16705 \dots < x < 2,

Increasing

: 2 < x < 3.37755 \dots,

Decreasing

: 3.37755 \dots < x < \infty

Plug the extreme point

x = - 3.54460 \dots

into

\frac{1 - x ^{3}}{x ^{2} - 4} \Rightarrow y = 5.31691 \dots

Minimum (- 3.54460 \dots, 5.31691 \dots)

Plug the extreme point

x = 0

into

\frac{1 - x ^{3}}{x ^{2} - 4} \Rightarrow y = - \frac{1}{4}

Maximum (0, - \frac{1}{4})

Plug the extreme point

x = 0.16705 \dots

into

\frac{1 - x ^{3}}{x ^{2} - 4} \Rightarrow y = - 0.25058 \dots

Minimum (0.16705 \dots, - 0.25058 \dots)

Plug the extreme point

x = 3.37755 \dots

into

\frac{1 - x ^{3}}{x ^{2} - 4} \Rightarrow y = - 5.06632 \dots

Maximum (3.37755 \dots, - 5.06632 \dots)

Minimum (- 3.54460 \dots, 5.31691 \dots), Maximum (0, - \frac{1}{4}), Minimum (0.16705 \dots, - 0.25058 \dots), Maximum (3.37755 \dots, - 5.06632 \dots)

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