Taking the derivative with e: f(x) = e^x - e^(7x)

[CaseyDaisy]

Hi, I am desperately trying to finish my math homework and this one problem keeps stumping me. Here's the problem:

Let f(x) = e^x - e^(7 x).

Find all extreme values (if any) of f on the interval 0 <= x <= 1. Determine at which numbers in the interval these values occur. Remember to check for endpoint extrema.

So I took the derivative and got f'(x)= e^x-7e^7x and set that equal to zero so that I could find my extreme values. However, I ran into some difficulty solving for x; if 0=e^x-7e^7x, how do I get x by itself? I cannot just move the 7e^7x to the other side and take the ln of both sides to get the exponents to equal each other because the 7 in front of the e and the 7 in the exponent. What do I do? I have tried factoring, but I cannot seem to get that to work either

 

[mmm4444bot]

Re: Taking the derivative with e

...

0 = e^x - 7e^(7x)

how do I get x by itself?

...

There is a non-zero quantity by which you could divide both sides of the equation to eliminate one of the two variables.

Cheers,

~ Mark

 

[fasteddie65]

f'(x)= e^x-7e^7x = e^x(1 - 7e^6x) = 0
1 - 7e^6x = 0
1 = 7e^6x
(1/7) = e^6x
ln (1/7) = - ln 7 = 6x
(-ln 7)/6 = x