Question

On a coordinate plane, a parabola opens up. It goes through (negative 8, negative 2), has a vertex at (negative 5, negative 6.5), goes through (negative 2, negative 2), and has a y-intercept at (0, 6).
Over which interval is the graph of f(x) = one-halfx2 + 5x + 6 increasing?

(–6.5, ∞)
(–5, ∞)
(–∞, –5)
(–∞, –6.5)

Answer 1

(-5, ∞)

Since f(x) = (1/2)x^2 + 5x + 6 has a = 1/2 > 0, its vertex at x = -5 is a minimum. Equivalently f'(x) = x + 5, which is positive for x > -5, so f is increasing on (-5, ∞).