Question

How many critical points does the function f(x)=((x −1)^6) ((x + 5)^7) have?

Answer 1

critical points are points in the domain where f'(x) = 0 or does not exist. Since this is a polynomial, we don't have to worry about the latter possibility.

f'(x) = 6(x-1)^5(x+5)^7 + 7(x-1)^6(x+5)^6
= (x-1)^5(x+5)^6 (6(x+5) + 7(x-1))
= (x-1)^5(x+5)^6 (13x + 23)

There are thus three critical points, at x=1,-5,-23/13