What is the critical 4e^{-2x}x^2-8e^{-2x}x+2e^{-2x} ?

The critical 4e^{-2x}x^2-8e^{-2x}x+2e^{-2x} is x=(3-sqrt(3))/2 ,x=(3+sqrt(3))/2

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critical 4e^{-2x}x^2-8e^{-2x}x+2e^{-2x}

critical points

4 e^{- 2 x} x^{2} - 8 e^{- 2 x} x + 2 e^{- 2 x}

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

Solution steps

4 e^{- 2 x} x^{2} - 8 e^{- 2 x} x + 2 e^{- 2 x}

Find where

f^{'} (x)

is equal to zero or undefined

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

Identify critical points not in

f (x)

domain

Domain of

4 e^{- 2 x} x^{2} - 8 e^{- 2 x} x + 2 e^{- 2 x} : - \infty < x < \infty

All critical points are in domain

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

f (x) = 12 x^{5} + 30 x^{4} - 300 x^{3} + 4

f (x) = x^{3} y + 12 x - 8 y

f (x) = - 4 x^{3} - 4 x^{2} + 5 x - 1

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

f (x, y) = x^{3} + y^{3} - 27 x - 12 y

What is the critical 4e^{-2x}x^2-8e^{-2x}x+2e^{-2x} ?
The critical 4e^{-2x}x^2-8e^{-2x}x+2e^{-2x} is x=(3-sqrt(3))/2 ,x=(3+sqrt(3))/2