Question

Discuss the curves y = x⁴ – 4x³ with respect to concavity and points of inflection.

Answer 1

f(x) = x⁴ – 4x³ 

⇒ f'(x) = 4x³ – 12x²

f”(x) = 12x² – 24x = 12x (x – 2) 

Since f”(x) = 0 when x = 0 or 2, we divide the real line into three intervals. 

(-∞, 0) , (0, 2), (2, ∞) and complete the following chart

The point (0, f(0) i.e., (0, 0) is an inflection point since the curve changes from concave upward to concave downward there. Also (2, f(2)) i.e., (2, -16) is an inflection point since the curve changes from concave downward to concave upward there.