How do I find all the critical numbers for this function?

Question

f(x)=(9-x^2)^(3/5)

how do i do the work for this problem and which answer is correct?

A) 0 B) 3 C) -3 and 3 D) -3, 0, and 3 E) none of the above

Answer 1

The critical numbers of a function are the values for which the derivative of the function is zero or undefined.

Use the chain rule to get the derivative:

f'(x) = 3/5(9 - x^2)^(-2/5) (-2x)

And then simplify a little:

f'(x) = -6x/(5(9 - x^2)^(2/5))

Setting it equal to zero:

-6x/(5(9 - x^2)^(2/5) = 0

There's only one value that makes the numerator zero: x = 0, so that's one critical point.

There are 2 valules that make the denominator zero, and therefore the derivative undefined: x = 3 and x = -3.

So I say the answer is D.

Answer 2

Are you in calculus or algebra?
If you're in calculus, the critical numbers are as Jim said, and so his answer is correct.
If yo'ure in algebra, the numbers for which the function itself is 0 or undefined are called the critical numbers in some texts, although this is not standard.