What is the extreme f(x)=x(x-1)^3 ?

The extreme f(x)=x(x-1)^3 is Minimum(1/4 ,-27/256),Saddle(1,0)

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extreme f(x)=x(x-1)^3

extreme points

f (x) = x (x - 1)^{3}

Minimum (\frac{1}{4}, - \frac{27}{256}), Saddle (1, 0)

Solution steps

f^{'} (x) = (x - 1)^{2} (4 x - 1)

Find intervals:

Decreasing

: - \infty < x < \frac{1}{4},

Increasing

: \frac{1}{4} < x < 1,

Increasing

: 1 < x < \infty

Plug

x = \frac{1}{4}

into

x (x - 1)^{3} : - \frac{27}{256}

Minimum (\frac{1}{4}, - \frac{27}{256})

Plug

x = 1

into

x (x - 1)^{3} : 0

Saddle (1, 0)

Minimum (\frac{1}{4}, - \frac{27}{256}), Saddle (1, 0)

What is the extreme f(x)=x(x-1)^3 ?
The extreme f(x)=x(x-1)^3 is Minimum(1/4 ,-27/256),Saddle(1,0)