Question

how do i find the critical points of f(x) = (x-2)^5 (x+3)^4?

Answer 1

How would you explain a critical point? A minimum/maximum? a zero?

Answer 2

hmm i dont know haha... i know that i need to find the derivative first though.

Answer 3

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,

Answer 4

So it sounds like you need to take the derivative, set it to zero and then solve for x.

Answer 5

and i got \[f'(x) = 5(x-2)^4(x+3)^4 + 4(x+3)^3(x-2)^5\]

Answer 6

embarrassing, but i honestly have no idea how to factor that. ha.

Answer 7

Well, the idea is to find an alternative to brute force.

Answer 8

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.