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Popular >

extreme f(x)=x^5-x^3+1

Solution

extreme points

f (x) = x^{5} - x^{3} + 1

Solution

Maximum (- \frac{3}{5}, - (\frac{3}{5})^{\frac{5}{2}} + (\frac{3}{5})^{\frac{3}{2}} + 1), Saddle (0, 1), Minimum (\frac{3}{5}, (\frac{3}{5})^{\frac{5}{2}} - (\frac{3}{5})^{\frac{3}{2}} + 1)

Solution steps

f^{'} (x) = 5 x^{4} - 3 x^{2}

Find intervals:

Increasing

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

Decreasing

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

Decreasing

: 0 < x < \frac{3}{5},

Increasing

: \frac{3}{5} < x < \infty

Plug

x = - \frac{3}{5}

into

x^{5} - x^{3} + 1 : - (\frac{3}{5})^{\frac{5}{2}} + (\frac{3}{5})^{\frac{3}{2}} + 1

Maximum (- \frac{3}{5}, - (\frac{3}{5})^{\frac{5}{2}} + (\frac{3}{5})^{\frac{3}{2}} + 1)

Plug

x = 0

into

x^{5} - x^{3} + 1 : 1

Saddle (0, 1)

Plug

x = \frac{3}{5}

into

x^{5} - x^{3} + 1 : (\frac{3}{5})^{\frac{5}{2}} - (\frac{3}{5})^{\frac{3}{2}} + 1

Minimum (\frac{3}{5}, (\frac{3}{5})^{\frac{5}{2}} - (\frac{3}{5})^{\frac{3}{2}} + 1)

Maximum (- \frac{3}{5}, - (\frac{3}{5})^{\frac{5}{2}} + (\frac{3}{5})^{\frac{3}{2}} + 1), Saddle (0, 1), Minimum (\frac{3}{5}, (\frac{3}{5})^{\frac{5}{2}} - (\frac{3}{5})^{\frac{3}{2}} + 1)

Popular Examples

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y = \frac{x}{x ^{2} + 16}

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f (x) = \frac{4}{3} x^{3} - \frac{13}{2} x^{2} - 3 x + 11

extreme f(x)=3x^3-14x+20x-8 extreme points

f (x) = 3 x^{3} - 14 x + 20 x - 8

extreme f(x)=(2x^2-5x)^3 extreme points

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Frequently Asked Questions (FAQ)

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