Let p(x)=x^2(x−a), where a is constant and a>0.
Find the local maxima and minima of p.
This is what I had initially done (which is incorrect):
I took the derivative and set it equal to 0. I found a in terms of x, and then plugged this value for a into the original function, so it would only be in terms of x. Then, I took the derivative again (still the original function, but only in terms of x now). I set it equal to zero and solved for x. I got x=0 as my only critical point. I tested the points -1 and 1 and found that 0 would be a local max.
So, this is definitely wrong. Can someone explain how to solve this problem?
Then, I need to explain:
What effect does increasing a have on the x-position of the maximum(s)?
On the minimum(s)?
What effect does increasing a have on the y-coordinate of the maximum(s)?
On the minimum(s)?
Thanks!