Consider the function g(x) = x^2sqrt(5-x)

a)find the intervals on which the function is increasing and decreasing.

Well, start by taking g'(x) by using the product rule:

g'(x) = 2xsqrt(5-x) + x^2*(1/2)(5-x)^(-1/2)*-1

g'(x) = 2x*sqrt(5-x) -(1/2)*x^2/sqrt(5-x) = 0

2x*sqrt(5-x) = x^2/(2*sqrt(5-x))

Multiply both sides by 2*sqrt(5-x) to get:

4x*(5-x) = x^2

20x - 4x^2 = x^2

5x^2 - 20x = 0

5x(x-4) = 0

x = 0, x = 4

Hope this helps.