What is the extreme f(x)=20x(1-x)^3 ?

The extreme f(x)=20x(1-x)^3 is Maximum(1/4 , 135/64),Saddle(1,0)

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extreme f(x)=20x(1-x)^3

extreme points

f (x) = 20 x (1 - x)^{3}

Maximum (\frac{1}{4}, \frac{135}{64}), Saddle (1, 0)

Solution steps

f^{'} (x) = 20 ((1 - x)^{3} - 3 x (1 - x)^{2})

Find intervals:

Increasing

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

Decreasing

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

Decreasing

: 1 < x < \infty

Plug

x = \frac{1}{4}

into

20 x (1 - x)^{3} : \frac{135}{64}

Maximum (\frac{1}{4}, \frac{135}{64})

Plug

x = 1

into

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

Saddle (1, 0)

Maximum (\frac{1}{4}, \frac{135}{64}), Saddle (1, 0)

4 x^{3} - 48 x - 1

f (x) = x^{3} - 9 x^{2} + 15 x + 8

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

f (x) = \frac{1}{2} x^{4} - 4 x^{2} + 6

f (x) = x^{3} - 9 x^{2} + 15 x + 5

What is the extreme f(x)=20x(1-x)^3 ?
The extreme f(x)=20x(1-x)^3 is Maximum(1/4 , 135/64),Saddle(1,0)