What is the extreme f(x)=x^5e^{2x} ?

The extreme f(x)=x^5e^{2x} is Minimum(-5/2 ,-(3125)/(32e^5)),Saddle(0,0)

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extreme f(x)=x^5e^{2x}

extreme points

f (x) = x^{5} e^{2 x}

Minimum (- \frac{5}{2}, - \frac{3125}{32 e ^{5}}), Saddle (0, 0)

Solution steps

f^{'} (x) = 5 x^{4} e^{2 x} + e^{2 x} \cdot 2 x^{5}

Find intervals:

Decreasing

: - \infty < x < - \frac{5}{2},

Increasing

: - \frac{5}{2} < x < 0,

Increasing

: 0 < x < \infty

Plug

x = - \frac{5}{2}

into

x^{5} e^{2 x} : - \frac{3125}{32 e ^{5}}

Minimum (- \frac{5}{2}, - \frac{3125}{32 e ^{5}})

Plug

x = 0

into

x^{5} e^{2 x} : 0

Saddle (0, 0)

Minimum (- \frac{5}{2}, - \frac{3125}{32 e ^{5}}), Saddle (0, 0)

h (t) = - 16 t^{2} + 96 t + 7

f (x, y) = 4 - y^{3} - x^{2} - 3 x y

- 74 + 30 x - 3 x^{2}

f (x, y) = 50 - 7 x^{2} - 2 y^{2}

4^{x} + 1

What is the extreme f(x)=x^5e^{2x} ?
The extreme f(x)=x^5e^{2x} is Minimum(-5/2 ,-(3125)/(32e^5)),Saddle(0,0)