For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it's decreasing.

y= x square root (9-x^2)

I did

y = x sqrt(9-x^2)

y' = (9-2x^2)/sqrt(9-x^2)

0 = y' = 9-2x^2

2x^2 = 9

x^2 = 9/2

x = sqrt(9/2) = 3/sqrt(2), -3/sqrt(2) (These are your critical numbers)

Now just do a sign chart for y'

-oo ---------- -3/sqrt(2) ------------ 3/sqrt(2) --------------- oo y'

- - - - - - - - - - - - 0 + + + + + + + + 0 - - - - - - - - - - -

The function is decreasing where it is - and increasing where it is +.

So Increasing from (-3/sqrt(2),3sqrt(2)) and decreasing from (-oo,-3/sqrt(2)) and (3/sqrt(2),oo)

A. sqrt(9/2) = 3/sqrt(2), -3/sqrt(2)

B. (-3/sqrt(2),3sqrt(2))

C. (-oo,-3/sqrt(2)) and (3/sqrt(2),oo)