### Home > CALC > Chapter 6 > Lesson 6.4.3 > Problem 6-163

If *g*(*x*) is a differentiable function such that *g*(*x*) < 0 for all real numbers *x* and if *f* ′(*x*) = (*x*^{2} − 9)*g*(*x*), determine if *f*(3) and *f*(−3) are relative minimums or maximums. Justify your answer. Homework Help ✎

Determine if *f*(3) is an extrema candidate.

That is, does *f* '(3) = 0 or does *f* '(3) = DNE*f* '(3) = (3²−9)(some negative value) = 0

Yes, *f*(3) is an extrema candidate! It could be a relative max, a relative min, or neither.

Determine if *f*(3) is a relative max, relative min or neither. Use the 1st-derivative test:

To do this, determine if *f* ′(*x*) changes sign at *x* = 3.

Evaluate a little to the left and a little to the right.

To the left: *f* ′(2.9) = (2.9³ − 9)(some negative value) = (negative)(negative) = positive.

To the right: *f* ′(3.1) = (3.1³ − 9)(some negative value) = (positive)(negative) = negative.

Conclusion: Since *f* ′(3) = 0 and *f* ′(*x*) changes from positive to negative at *x* = 3, therefore *f*(3) is a relative maxima.

Now test *f*(−3).