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Popular >

extreme (x^{1/3})/(x^2+8)-10

Solution

extreme points

\frac{x ^{\frac{1}{3}}}{x ^{2} + 8} - 10

Solution

Minimum (- \frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}}, - \frac{5 ^{\frac{5}{6}}}{24 \cdot 2 ^{\frac{1}{2}}} - 10), Maximum (\frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}}, \frac{5 ^{\frac{5}{6}}}{24 \cdot 2 ^{\frac{1}{2}}} - 10)

Solution steps

f^{'} (x) = - \frac{( 5 ^{\frac{1}{2}} x + 2 \cdot 2 ^{\frac{1}{2}} ) ( 5 ^{\frac{1}{2}} x - 2 \cdot 2 ^{\frac{1}{2}} )}{3 x ^{\frac{2}{3}} ( x ^{2} + 8 ) ^{2}}

Find intervals:

Decreasing

: - \infty < x < - \frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}},

Increasing

: - \frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}} < x < 0,

Increasing

: 0 < x < \frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}},

Decreasing

: \frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}} < x < \infty

Plug

x = - \frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}}

into

\frac{x ^{\frac{1}{3}}}{x ^{2} + 8} - 10 : - \frac{5 ^{\frac{5}{6}}}{24 \cdot 2 ^{\frac{1}{2}}} - 10

Minimum (- \frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}}, - \frac{5 ^{\frac{5}{6}}}{24 \cdot 2 ^{\frac{1}{2}}} - 10)

Plug

x = \frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}}

into

\frac{x ^{\frac{1}{3}}}{x ^{2} + 8} - 10 : \frac{5 ^{\frac{5}{6}}}{24 \cdot 2 ^{\frac{1}{2}}} - 10

Maximum (\frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}}, \frac{5 ^{\frac{5}{6}}}{24 \cdot 2 ^{\frac{1}{2}}} - 10)

Minimum (- \frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}}, - \frac{5 ^{\frac{5}{6}}}{24 \cdot 2 ^{\frac{1}{2}}} - 10), Maximum (\frac{2 ^{\frac{3}{2}}}{5 ^{\frac{1}{2}}}, \frac{5 ^{\frac{5}{6}}}{24 \cdot 2 ^{\frac{1}{2}}} - 10)

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Frequently Asked Questions (FAQ)

What is the extreme (x^{1/3})/(x^2+8)-10 ?
The extreme (x^{1/3})/(x^2+8)-10 is