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- #1

The question is:

What are the intervals where function f(x)=e

^{2x}-2e

^{x}is concave and convex respectively.

I have derived f(x) to get

f'(x)=2e

^{2x}-2e

^{x}which I derived again to get the second derative:

f''(x) = 4e

^{2x}-2e

^{x}

After which I put up the equation

f''(x) = 0, since this determines the inflection point

which becomes 4e

^{2x}-2e

^{x}=0

This is where I get stuck. I know have to find the value of x, and then test values bigger and smaller of it using the equations for concave/convex parabolas, and I'm pretty sure that the answer is x=ln 0.5 but I just don't seem to be able to show it.. Logarithms was never my thing it appears.

Either way, I tried doing this

2(2e

^{2x}-e

^{x})=0

2e

^{2x}-e

^{x}=0

2e

^{2x=}e

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

And this is where I don't know how to continue..