Question

How many integer values of a are there such that
f(x)=x^+ax^2+8ax+25
has no local extrema?

Answer 1

Assuming you meant

x^3+ax^2+8ax+25

we want a derivative with no zeros. So,

f' = 3x^2 + 2ax + 8a
f" = 6x+2a

this will have no zeros if the discriminant is negative, so we need

(2a)^2 - 4(3)(8a) < 0
4a^2 - 96a < 0
4a(a-24) < 0

So 0<a<24

At a=0 and a=24, f"(0) when f'=0, so there's an inflection point, so no extrema.

So, 0 <= a <= 24