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March 1, 2015

Posted by **Abigail** on Tuesday, June 15, 2010 at 10:30am.

- Calculus -
**MathMate**, Tuesday, June 15, 2010 at 7:04pmLet

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

the domain of f(x) is (-∞,∞).

Thus the extrema of f(x) can be found at point(s) where f'(x)=0.

f'(x)=e^{x}- 2e^{2x}

and f'(x)=0 when

e^{x}= 2e^{2x}

2e^{x}=1

x=ln(1/2) (only root)

Since f"(x)=e^{x}- 4e^{2x}

and

f"(-ln(1/2)) = -1/2

we conclude that a maximum exists at x=ln(1/2) since f" is negative.

Can you take it from here?

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