Calculus of a Single Variable, Chapter 3, 3.2, Section 3.2, Problem 7

You need to find the two x intercepts of the function, hence, you need to solve for x the equation f(x) = 0, such that:
f(x) = x*sqrt(x+4) = 0
x*sqrt(x+4) = 0 x = 0
sqrt(x+4) = 0 => x + 4 = 0^2 => x + 4 = 0 => x = -4
You need to evaluate the derivative of the function, using the product rule:
f'(x) = x'*sqrt(x+4) + x*(sqrt(x+4))' => f'(x) = sqrt(x+4) + x/(2sqrt(x+4))
You need to solve for x the equation f'(x) = 0:
sqrt(x+4) + x/(2sqrt(x+4)) = 0 => (2(x+4) + x)/(2sqrt(x+4)) = 0
2x + 8 + x = 0 => 3x = -8 => x = -8/3 = -2.(6)
Notice that -2.(6) is found between x intercepts -4 and 0.
Hence, the derivative of the function cancels at x = -2.(6), which is found between the x intercepts -4 and 0.