What is the extreme y=(x*(1+x)*(1-x))/(1+2x) ?

The extreme y=(x*(1+x)*(1-x))/(1+2x) is Maximum(0.45541…,0.18890…)

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extreme y=(x(1+x)(1-x))/(1+2x)

extreme points

y = \frac{x \cdot ( 1 + x ) \cdot ( 1 - x )}{1 + 2 x}

Maximum (0.45541 \dots, 0.18890 \dots)

Solution steps

Find the critical points:

x \approx 0.45541 \dots

Domain of

\frac{x \cdot ( 1 + x ) \cdot ( 1 - x )}{1 + 2 x} : x < - \frac{1}{2} or x > - \frac{1}{2}

Find intervals:

Increasing

: - \infty < x < - \frac{1}{2},

Increasing

: - \frac{1}{2} < x < 0.45541 \dots,

Decreasing

: 0.45541 \dots < x < \infty

Plug the extreme point

x = 0.45541 \dots

into

\frac{x \cdot ( 1 + x ) \cdot ( 1 - x )}{1 + 2 x} \Rightarrow y = 0.18890 \dots

Maximum (0.45541 \dots, 0.18890 \dots)

f (x) = \frac{( x ^{2} )}{x ^{2} - 81}

f (x) = 3 x^{3} - 12 x^{2} + 12 x - 6, 0 \leq x \leq 4

y = 3 x^{3} (x - 5)^{2}

y = \frac{1}{x - 8} - 9

y = x + \frac{25}{x}

What is the extreme y=(x*(1+x)*(1-x))/(1+2x) ?
The extreme y=(x*(1+x)*(1-x))/(1+2x) is Maximum(0.45541…,0.18890…)