What is the extreme f(x)=xsqrt(5-x^2) ?

The extreme f(x)=xsqrt(5-x^2) is

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extreme f(x)=xsqrt(5-x^2)

extreme

Maximum (- 5, 0), Minimum (- \frac{5}{2}, - \frac{5}{2}), Maximum (\frac{5}{2}, \frac{5}{2}), Minimum (5, 0)

Solution steps

Find the critical points:

x = - 5, x = - \frac{5}{2}, x = \frac{5}{2}, x = 5

Domain of

x 5 - x^{2} : - 5 \leq x \leq 5

Find intervals:

Decreasing

: - 5 < x < - \frac{5}{2},

Increasing

: - \frac{5}{2} < x < \frac{5}{2},

Decreasing

: \frac{5}{2} < x < 5

Plug the extreme point

x = - 5

into

x 5 - x^{2} \Rightarrow y = 0

Maximum (- 5, 0)

Plug the extreme point

x = - \frac{5}{2}

into

x 5 - x^{2} \Rightarrow y = - \frac{5}{2}

Minimum (- \frac{5}{2}, - \frac{5}{2})

Plug the extreme point

x = \frac{5}{2}

into

x 5 - x^{2} \Rightarrow y = \frac{5}{2}

Maximum (\frac{5}{2}, \frac{5}{2})

Plug the extreme point

x = 5

into

x 5 - x^{2} \Rightarrow y = 0

Minimum (5, 0)

Maximum (- 5, 0), Minimum (- \frac{5}{2}, - \frac{5}{2}), Maximum (\frac{5}{2}, \frac{5}{2}), Minimum (5, 0)