What is the extreme sqrt(x)ln(3x) ?

The extreme sqrt(x)ln(3x) is Minimum(1/(3e^2),-2/(sqrt(3)e))

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extreme sqrt(x)ln(3x)

extreme points

x ln (3 x)

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

Solution steps

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

Find intervals:

Decreasing

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

Increasing

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

Plug

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

into

x ln (3 x) : - \frac{2}{3 e}

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

f (x) = 3 - 7 x^{2}

f (x) = 12 x^{2} - 48 x + 32

f (x) = \frac{12}{7} x^{7} - 60 x^{5} + 576 x^{3}

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

f (x, y) = e^{5 x^{2} + 5 y^{2} + 3}

What is the extreme sqrt(x)ln(3x) ?
The extreme sqrt(x)ln(3x) is Minimum(1/(3e^2),-2/(sqrt(3)e))