Here's the question you clicked on:
ksandoval
how do i find the critical points of f(x) = (x-2)^5 (x+3)^4?
How would you explain a critical point? A minimum/maximum? a zero?
hmm i dont know haha... i know that i need to find the derivative first though.
A critical point of a function of a single real variable, ƒ(x), is a value x0 in the domain of ƒ where either the function is not differentiable or its derivative is 0,
So it sounds like you need to take the derivative, set it to zero and then solve for x.
and i got \[f'(x) = 5(x-2)^4(x+3)^4 + 4(x+3)^3(x-2)^5\]
embarrassing, but i honestly have no idea how to factor that. ha.
Well, the idea is to find an alternative to brute force.
Ok, when things look ugly like this, substitute something easy. Like A= x-2 and B=x+3 This gives you \[f'(X) = 5A^4B^4 +4B^3A^5\] This is easy to factor and set to zero. Then once you have solved for when THIS is zero with respect to A and B then resubstitute the A=x-2 and B=x+3 to get you final answer.