Question

How many critical points does the function:

(x+2)^5 * (x-3)^4

have?

How would I solve this quickly? The answer says something about an effect of multiplicity of the zeros of the function?

Answer 1

A critical point of a function within its domain is any point which is not differentiable or when its derivative is zero.

Since
f(x)=(x+2)^5 * (x-3)^4
is a polynomial, its domain is (-∞,∞) and differentiable throughout.
The only critical points are when the derivative is zero.
f'(x)=0 =>
4(x-3)^3(x+2)^5+5(x-3)^4(x+2)^4=0
which factors to:
(x-3)^3*(x+2)^4*(9*x-7)=0
We see that
x=3 (multiplicity 3)
x=-2 (multiplicity 4)
and
x=7/9
So there is a total of 3+4+1 = 8 critical points, out of which there are 3 distinct points.