Help with calculus problem with logarithm base e

[linapril]

I'm at one of those annoying stages where you know what the answer is, but you just can't seem to prove it... Would really appreciate some help with this one!

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..

 

[MarkFL]

Re: Help with calculus problem with logarithm base e!

You have correctly differentiated to get the second derivative:

$$f''(x)=4e^{2x}-2e^x$$

You factored, but you didn't factor completely. Consider:

$$f''(x)=2e^x\left(2e^x-1 \right)$$

Now equate that to zero to determine the critical value. which you are correct about, but you can show it using the above form. What do you find?

 

[linapril]

Re: Help with calculus problem with logarithm base e!

I'm not quite sure I understand... Why does 2e^x times 2e^x equal 4e^2x?
And also, If I weren't to factorise I would continue with
4e^2x-2e^x=0
4e^2x=2e^x
Is it incorrect to simplify this to
2e^2x=ex
And if so, why?

 

[MarkFL]

Re: Help with calculus problem with logarithm base e!

I'm not quite sure I understand... Why does 2e^x times 2e^x equal 4e^2x?
And also, If I weren't to factorise I would continue with
4e^2x-2e^x=0
4e^2x=2e^x
Is it incorrect to simplify this to
2e^2x=ex
And if so, why?

Recall the property of exponents:

$$a^b\cdot a^c=a^{b+c}$$

You could continue as you did to get:

$$4e^{2x}=2e^x$$

Now, what happens if you multiply through by $$\frac{e^{-x}}{2}$$? Are you in danger of multiplying through by zero in doing so? What does the above property of exponents yield?

 

[Deveno]

I'm at one of those annoying stages where you know what the answer is, but you just can't seem to prove it... Would really appreciate some help with this one!

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..

I think you have a mistake in going from this line:

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

to this line:

$2x \ln(2e) = x\ln(e)$

If we take logarithms of both sides, we get:

$\ln(2e^{2x}) = \ln(e^x)$

the right-hand side is $x$ since $\ln$ and $e^{(\_)}$ are inverse functions.

The left-hand side is:

$\ln(2\cdot e^{2x}) = \ln(2) + \ln(e^{2x}) = \ln(2) + 2x$

So your equation becomes:

$\ln(2) + 2x = x$

Which should be easy to solve for $x$.

 

[linapril]

Ahaaa, I get it now! Thank you so much!