Question

Identify the open intervals) where the function f of F(x)= x√10-x^2 is increasing.

Answer 1

I assume you meant
f(x) = x √(10 - x^2) and not the way you typed it

Of course the function would only be defined for

-√10 ≤ x ≤ √10

f' (x) = x (1/2)(10-x^2)^(-1/2) (-2x ) + (10-x^2)^(1/2)
= (10 - x^2)^(-1/2) [ -x^2 + 10-x^2]
= (10-2x^2)/√(10-x^2)

now the bottom is positive for our domain of x
so let's look only at the numerator

to be increasing the F' (x) has to be positive
so when is 10 - 2x^2 > 0

- 2x^2 > -10
2x^2 < 10
± x < √5
x < √5 and x ≥ -√5

the function increases for -√5 < x ≤ √5

confirmation:

http://www.wolframalpha.com/input/?i=plot+y+%3D+x√%2810-x%5E2%29