Calculus of a Single Variable, Chapter 3, 3.2, Section 3.2, Problem 6

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^2 + 6x = 0
You need to factor out x, such that:
x(x + 6) = 0 => x = 0
x + 6 = 0 => x = -6
You need to evaluate the derivative of the function:
f'(x) = (x^2 + 6x)' => f'(x) = 2x + 6
You need to solve for x the equation f'(x) = 0:
2x + 6 = 0 => 2x = -6 => x = -3
Notice that -3 is found between x intercepts -6 and 0.
Hence, the derivative of the function cancels at x = -3, which is found between the x intercepts -6 and 0.