Your work appears correct and a graph confirms your answer. x*sqrt(9-x^2) - Wolfram|Alpha
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)
Your work appears correct and a graph confirms your answer. x*sqrt(9-x^2) - Wolfram|Alpha