What is the extreme f(x)=x(ln(x))^2 ?

The extreme f(x)=x(ln(x))^2 is Maximum(1/(e^2), 4/(e^2)),Minimum(1,0)

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extreme f(x)=x(ln(x))^2

extreme points

f (x) = x (ln (x))^{2}

Maximum (\frac{1}{e ^{2}}, \frac{4}{e ^{2}}), Minimum (1, 0)

Solution steps

f^{'} (x) = ln^{2} (x) + 2 ln (x)

Find intervals:

Increasing

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

Decreasing

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

Increasing

: 1 < x < \infty

Plug

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

into

x ln^{2} (x) : \frac{4}{e ^{2}}

Maximum (\frac{1}{e ^{2}}, \frac{4}{e ^{2}})

Plug

x = 1

into

x ln^{2} (x) : 0

Minimum (1, 0)

Maximum (\frac{1}{e ^{2}}, \frac{4}{e ^{2}}), Minimum (1, 0)

f (x) = x^{3} + 3 x^{2} + 4

f (x, y) = x^{2} + x y + y^{2} + 4 y

f (x) = \frac{x}{9 + x ^{2}}

f (x, y) = x^{2} + y^{3} - 6 x y + 3 x + 6 y - 2

What is the extreme f(x)=x(ln(x))^2 ?
The extreme f(x)=x(ln(x))^2 is Maximum(1/(e^2), 4/(e^2)),Minimum(1,0)