What is the extreme f(x)=x-1/6 x^2-2/3 ln(x) ?

The extreme f(x)=x-1/6 x^2-2/3 ln(x) is Minimum(1, 5/6),Maximum(2, 4/3-2/3 ln(2))

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extreme f(x)=x-1/6 x^2-2/3 ln(x)

extreme points

f (x) = x - \frac{1}{6} x^{2} - \frac{2}{3} ln (x)

Minimum (1, \frac{5}{6}), Maximum (2, \frac{4}{3} - \frac{2}{3} ln (2))

Solution steps

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

Find intervals:

Decreasing

: 0 < x < 1,

Increasing

: 1 < x < 2,

Decreasing

: 2 < x < \infty

Plug

x = 1

into

x - \frac{1}{6} x^{2} - \frac{2}{3} ln (x) : \frac{5}{6}

Minimum (1, \frac{5}{6})

Plug

x = 2

into

x - \frac{1}{6} x^{2} - \frac{2}{3} ln (x) : \frac{4}{3} - \frac{2}{3} ln (2)

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

Minimum (1, \frac{5}{6}), Maximum (2, \frac{4}{3} - \frac{2}{3} ln (2))

What is the extreme f(x)=x-1/6 x^2-2/3 ln(x) ?
The extreme f(x)=x-1/6 x^2-2/3 ln(x) is Minimum(1, 5/6),Maximum(2, 4/3-2/3 ln(2))