Find intervals of increase or decrease in interval notation

Question

of the function x√(2-x)

Answer 1

f(x) = x √(2-x) Domain: 2-x > 0. So, x < 2

f'(x) = √(2-x) - x(1/2)(2-x)^-1/2 = √(2-x) - x / (2√2-x) = [2(2-x) - x] / (2√2-x) = (4-3x) / [2√(2-x)]

f'(x) = 0 when x = 4/3

When x < 4/3, f'(x) > 0. So, f(x) is increasing

When 4/3 < x < 2, f'(x) < 0. So, f is decreasing