What is the extreme f(x)=x^7e^{2x}+2 ?

The extreme f(x)=x^7e^{2x}+2 is Minimum(-7/2 ,-(823543)/(128e^7)+2),Saddle(0,2)

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extreme f(x)=x^7e^{2x}+2

extreme

Minimum (- \frac{7}{2}, - \frac{823543}{128 e ^{7}} + 2), Saddle (0, 2)

Solution steps

f^{'} (x) = 2 e^{2 x} x^{7} + 7 e^{2 x} x^{6}

Find intervals:

Decreasing

: - \infty < x < - \frac{7}{2},

Increasing

: - \frac{7}{2} < x < 0,

Increasing

: 0 < x < \infty

Plug

x = - \frac{7}{2}

into

x^{7} e^{2 x} + 2 : - \frac{823543}{128 e ^{7}} + 2

Minimum (- \frac{7}{2}, - \frac{823543}{128 e ^{7}} + 2)

Plug

x = 0

into

x^{7} e^{2 x} + 2 : 2

Saddle (0, 2)

Minimum (- \frac{7}{2}, - \frac{823543}{128 e ^{7}} + 2), Saddle (0, 2)

What is the extreme f(x)=x^7e^{2x}+2 ?
The extreme f(x)=x^7e^{2x}+2 is Minimum(-7/2 ,-(823543)/(128e^7)+2),Saddle(0,2)