What is the extreme f(x)=x(ln(x))^5 ?

The extreme f(x)=x(ln(x))^5 is Minimum(1/(e^5),-(3125)/(e^5)),Saddle(1,0)

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extreme f(x)=x(ln(x))^5

extreme

Minimum (\frac{1}{e ^{5}}, - \frac{3125}{e ^{5}}), Saddle (1, 0)

Solution steps

f^{'} (x) = ln^{5} (x) + 5 ln^{4} (x)

Find intervals:

Decreasing

: 0 < x < \frac{1}{e ^{5}},

Increasing

: \frac{1}{e ^{5}} < x < 1,

Increasing

: 1 < x < \infty

Plug

x = \frac{1}{e ^{5}}

into

x ln^{5} (x) : - \frac{3125}{e ^{5}}

Minimum (\frac{1}{e ^{5}}, - \frac{3125}{e ^{5}})

Plug

x = 1

into

x ln^{5} (x) : 0

Saddle (1, 0)

Minimum (\frac{1}{e ^{5}}, - \frac{3125}{e ^{5}}), Saddle (1, 0)

What is the extreme f(x)=x(ln(x))^5 ?
The extreme f(x)=x(ln(x))^5 is Minimum(1/(e^5),-(3125)/(e^5)),Saddle(1,0)