y = f (x) = 2x3 + 5x2 – 4x
f'(x) = 6x2 + 10x – 4
f”(x) = 12x + 10; f'” (x) = 12
f”(x) = 0 ⇒ 12x + 10 = 0
Consider x in (-∞, -5/6) say x = -1
f”(x)= -12 + 10 = -2 < 0 ⇒ the curve convex upward in the interval (-∞, -5/6) Consider x in (-5/6, ∞) say x = 0 f”(x) = 0 + 10 = 10 > 0
⇒ the curve is concave upward in (-5/6, -∞)
Thus, the curve is concave upward in (-5/6, ∞) and convex upward in (-∞, – 5/6)
The point of inflection is ((-5/6), (305/54))