Posted by **Mira** on Friday, April 22, 2011 at 11:03pm.

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?

- calculus -
**MathMate**, Friday, April 22, 2011 at 11:15pm
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.

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