Logarithmic/Exponential Optimization

[VyseV1]

Im having just a bit of bother with this kind of optimization. I just can't seem to get the answers to work out :/ For example:

e^-x - e^-3x on 0<=x<=10

Well i know the derivative of that is -e^-x + 3e^-3x but for that point onwards im a little trapped. I know i need to find the critical point but i cant seem to get to it.

Another question that has me in some worry is:
10^6(1+(x-1)e^-.001x)
Between 0<=x<=2000

that one has me clueless.

 

[galactus]

\(\displaystyle \L\\3e^{-3x}-e^{-x}=0\)

What you could do is let \(\displaystyle \L\\u=e^{-x}\)

Then you have \(\displaystyle \L\\3u^{3}-u=0\)

\(\displaystyle \L\\u(3u^{2}-1)=0\)

Can you solve now. Remember, u=e^-x.

 

[VyseV1]

Ah yes i see the factoring now, which means the inside bracket shows the critical point, i see.

So afterwardxs would you take the ln of both sides to get the exponent by itself?

 

[galactus]

Yep.

\(\displaystyle \L\\u=\frac{1}{\sqrt{3}}\)

Now, since \(\displaystyle \L\\u=e^{-x}\), find x.