Posted by **andy** on Thursday, July 18, 2013 at 12:40pm.

How many integer values of a are there such that

f(x)=x^+ax^2+8ax+25

has no local extrema?

- calculus -
**Steve**, Thursday, July 18, 2013 at 1:10pm
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

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