The derivative of is
Therefore the derivative of is
To find the sign you have to remember that is always strictly positive
Therefore the sign of the derivative of is the same as the sign of
Now I think that you can finish the job !
I have no problem finding maximum and minimums of functions, except when there's an e involved. For example finding the max and min values of e^((x^3)-x) on the closed interval -1 and 0. First off I don't know how to find the derivative of a power raised to another power, and even when I cheat and use a derivative calculator to find that, I don't know how to set the remaining derivative with e equal to 0. Would someone mind helping me out here?
You are studying the function on interval [-1,0]
The derivative of is
To find the max and min of the function you have to solve on [-1,0]
This is equivalent to solve on [-1,0]
On [-1,0] you can find only one value for which
This value is
Now you have to find the sign of the derivative on [-1,0] in order to know where the function is increasing and decreasing
You seem to be misunderstanding. I don't need to find where the function is increasing or decreasing, I just have to find the absolute min and absolute max values of the function. That's it. Because it's absolute min/max and not just relative min/max, I plugged in the roots -1 and 0, which gave me 1, which I tried for both the min and max answer, and it turned out it is indeed the min. Now when I solve 3x^2-1=0, the resulting fraction that you provided, and I already tried, gives a number LOWER than the min when plugged into the function so it CANNOT be the max. That's precisely why I'm confused about this particular problem.
Oh jeez. I kept plugging in only POSITIVE sqrt 3/3, rather than negative. My apologies. This leads me to another similar question, however. What would you do if you have a similar function with e involved, and are asked to find the same thing (absolute min/max values) but the exponent's derivative no longer contains an x.
F(x) = e^-x - e^-2x
When you pull the exponents out in front there are no longer x's attached so how would you set it equal to 0 if e to any power can never equal 0.