Concavity & Inflection Points

Business Calculus · Axiom Academy

LESSON Concavity & Inflection Points Understanding the shape of curves Concavity describes how a curve bends. Think of it as the curve's "cupping" direction. Opens upward like a cup. The slope is increasing . Opens downward like a frown. The slope is decreasing . Imagine driving along the curve. If you're turning left (steering wheel turned left), the curve is concave up. If you're turning right, it's concave down. The second derivative f''(x) measures how the slope is changing. If f''(x) > 0: The slope f'(x) is increasing → Concave up If f''(x) decreasing → Concave down On the concave up part (left), the tangent lines get steeper as we move right (slope increases) On the concave down part (right), the tangent lines get less steep (slope decreases) An inflection point is where a function changes concavity—from concave up to concave down, or vice versa. f''(c) = 0 is necessary but NOT sufficient. The concavity must actually change at c for it to be an inflection point. Always verify by checking the sign of f'' on both sides. Example: Finding Inflection Points Step 3: Verify concavity changes For x < 2: f''(1) = 6(1) - 12 = -6 < 0 (concave down) For x > 2: f''(3) = 6(3) - 12 = 6 > 0 (concave up) Since concavity changes from down to up at x = 2, this is an inflection point. The inflection point is (2, f(2)) = (2, 3) Business Meaning of Inflection Points Concave up → Concave down Growth is accelerating → Growth is decelerating The inflection point marks "peak acceleration"

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