  ### Home > APCALC > Chapter 6 > Lesson 6.4.3 > Problem6-156

6-156.

Let $g$ be a differentiable function such that $g(x) < 0$ for all $x$ and $f^\prime(x) = (x^2 - 9)g(x)$. Determine if $f (3)$ and $f (–3)$ are relative extrema. Justify your answer.

Determine if $f(3)$ is an extrema candidate.
That is, does $f^\prime(3) = 0$ or does $f ^\prime(3) =$ DNE

$f'(3)=(3^2−9)$(some negative value) $= 0$
Yes, $f\left(3\right)$ is an extrema candidate! It could be a relative max, a relative min, or neither.

Determine if $f\left(3\right)$ is a relative max, relative min or neither. Use the 1st-derivative test:
To do this, determine if $f ^\prime(x)$ changes sign at $x = 3$.
Evaluate a little to the left and a little to the right.
To the left: $f^\prime(2.9)=(2.9^3−9)$(some negative value) $=$ (negative)(negative) $=$ positive.
To the right: $f^\prime(3.1) = (3.1^3 − 9)$(some negative value) $=$ (positive)(negative) $=$ negative.
Conclusion: Since $f^\prime(3) = 0$ and $f ^\prime(x)$ changes from positive to negative at $x = 3$, therefore $f(3)$ is a relative maxima.

Now test $f(−3)$.