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Popular >

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

Solution

extreme points

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

Solution

Minimum \frac{3 - 7}{2}, \frac{- 2 ln ( \frac{3 - 7}{2} ) + 10 - 3 7}{4}, Maximum \frac{3 + 7}{2}, \frac{- 2 ln ( \frac{3 + 7}{2} ) + 10 + 3 7}{4}

Solution steps

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

Find intervals:

Decreasing

: 0 < x < \frac{3 - 7}{2},

Increasing

: \frac{3 - 7}{2} < x < \frac{3 + 7}{2},

Decreasing

: \frac{3 + 7}{2} < x < \infty

Plug

x = \frac{3 - 7}{2}

into

3 x - \frac{1}{2} x^{2} - \frac{1}{2} ln (x) : \frac{- 2 ln ( \frac{3 - 7}{2} ) + 10 - 3 7}{4}

Minimum \frac{3 - 7}{2}, \frac{- 2 ln ( \frac{3 - 7}{2} ) + 10 - 3 7}{4}

Plug

x = \frac{3 + 7}{2}

into

3 x - \frac{1}{2} x^{2} - \frac{1}{2} ln (x) : \frac{- 2 ln ( \frac{3 + 7}{2} ) + 10 + 3 7}{4}

Maximum \frac{3 + 7}{2}, \frac{- 2 ln ( \frac{3 + 7}{2} ) + 10 + 3 7}{4}

Minimum \frac{3 - 7}{2}, \frac{- 2 ln ( \frac{3 - 7}{2} ) + 10 - 3 7}{4}, Maximum \frac{3 + 7}{2}, \frac{- 2 ln ( \frac{3 + 7}{2} ) + 10 + 3 7}{4}

Popular Examples

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Frequently Asked Questions (FAQ)

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