The Second Derivative and Concavity

Calculus 1 · Axiom Academy

LESSON The Second Derivative and Concavity How f''(x) reveals the way a curve bends — cupping up, capping down, and the inflection point where it switches. 1. Concave Up: the slope keeps increasing Watch a point slide left to right along a curve that opens upward like a cup . The tangent line is glued to it. Notice the tangent rotates counter-clockwise — it starts tilted steeply downhill and ends tilted steeply uphill. The slope is climbing the whole way , and "slope of the slope" climbing is exactly . The slope f'(x) is increasing, so its derivative f''(x) is positive 2. Concave Down: the slope keeps decreasing Now the same point slides along a curve that caps over like a hill . This time the tangent rotates clockwise — steeply uphill at the start, flat at the crest, steeply downhill at the end. The slope is falling the whole way , so the rate of change of the slope is negative: . If on an interval, f is concave up there (slope increasing). If on an interval, f is concave down there (slope decreasing). If f''(x) = 0 and changes sign , that point may be an inflection point — concavity flips. 3. The Inflection Point: where the bending flips An inflection point is where a curve switches from concave down to concave up (or the reverse). The classic case is f(x) = x^3 . Differentiate twice: f'(x) = 3x^2 and f''(x) = 6x . That second derivative is negative for and positive for — it changes sign right at x = 0 .

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