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Popular >

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

Solution

extreme points

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

Solution

Maximum (\frac{- 3 - 5}{2}, \frac{e ^{- 3 - 5} ( 1 + 3 5 )}{2} - 5 e^{- 3 - 5}), Minimum (\frac{- 3 + 5}{2}, \frac{e ^{- 3 + 5} ( 1 - 3 5 )}{2} + 5 e^{- 3 + 5})

Solution steps

f^{'} (x) = 2 e^{2 x} (x^{2} + 3 x + 1)

Find intervals:

Increasing

: - \infty < x < \frac{- 3 - 5}{2},

Decreasing

: \frac{- 3 - 5}{2} < x < \frac{- 3 + 5}{2},

Increasing

: \frac{- 3 + 5}{2} < x < \infty

Plug

x = \frac{- 3 - 5}{2}

into

e^{2 x} (x^{2} + 2 x) : \frac{e ^{- 3 - 5} ( 1 + 3 5 )}{2} - 5 e^{- 3 - 5}

Maximum (\frac{- 3 - 5}{2}, \frac{e ^{- 3 - 5} ( 1 + 3 5 )}{2} - 5 e^{- 3 - 5})

Plug

x = \frac{- 3 + 5}{2}

into

e^{2 x} (x^{2} + 2 x) : \frac{e ^{- 3 + 5} ( 1 - 3 5 )}{2} + 5 e^{- 3 + 5}

Minimum (\frac{- 3 + 5}{2}, \frac{e ^{- 3 + 5} ( 1 - 3 5 )}{2} + 5 e^{- 3 + 5})

Maximum (\frac{- 3 - 5}{2}, \frac{e ^{- 3 - 5} ( 1 + 3 5 )}{2} - 5 e^{- 3 - 5}), Minimum (\frac{- 3 + 5}{2}, \frac{e ^{- 3 + 5} ( 1 - 3 5 )}{2} + 5 e^{- 3 + 5})

Popular Examples

extreme f(x,y)=2x^2-4x+3y^2+7 extreme points

f (x, y) = 2 x^{2} - 4 x + 3 y^{2} + 7

extreme f(x)=x^2+3xy+y^2 extreme points

f (x) = x^{2} + 3 x y + y^{2}

extreme f(x)=(7x)/(x^2+4)extreme points

f (x) = \frac{7 x}{x ^{2} + 4}

extreme f(x)=-3x^4-2x^3+9x^2 extreme points

f (x) = - 3 x^{4} - 2 x^{3} + 9 x^{2}

extreme y= x/(x^2+49)extreme points

y = \frac{x}{x ^{2} + 49}

Frequently Asked Questions (FAQ)

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