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Popular >

extreme f(x)=(x^4)/4-(x/3)^3-x^2+1

Solution

extreme points

f (x) = \frac{x ^{4}}{4} - (\frac{x}{3})^{3} - x^{2} + 1

Solution

Minimum (\frac{1 - 649}{18}, \frac{649 649 - 158437}{157464} + 1), Maximum (0, 1), Minimum (\frac{1 + 649}{18}, - \frac{158437 + 649 649}{157464} + 1)

Solution steps

f^{'} (x) = x^{3} - \frac{x ^{2}}{9} - 2 x

Find intervals:

Decreasing

: - \infty < x < \frac{1 - 649}{18},

Increasing

: \frac{1 - 649}{18} < x < 0,

Decreasing

: 0 < x < \frac{1 + 649}{18},

Increasing

: \frac{1 + 649}{18} < x < \infty

Plug

x = \frac{1 - 649}{18}

into

\frac{x ^{4}}{4} - (\frac{x}{3})^{3} - x^{2} + 1 : \frac{649 649 - 158437}{157464} + 1

Minimum (\frac{1 - 649}{18}, \frac{649 649 - 158437}{157464} + 1)

Plug

x = 0

into

\frac{x ^{4}}{4} - (\frac{x}{3})^{3} - x^{2} + 1 : 1

Maximum (0, 1)

Plug

x = \frac{1 + 649}{18}

into

\frac{x ^{4}}{4} - (\frac{x}{3})^{3} - x^{2} + 1 : - \frac{158437 + 649 649}{157464} + 1

Minimum (\frac{1 + 649}{18}, - \frac{158437 + 649 649}{157464} + 1)

Minimum (\frac{1 - 649}{18}, \frac{649 649 - 158437}{157464} + 1), Maximum (0, 1), Minimum (\frac{1 + 649}{18}, - \frac{158437 + 649 649}{157464} + 1)

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Frequently Asked Questions (FAQ)

What is the extreme f(x)=(x^4)/4-(x/3)^3-x^2+1 ?
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