Second Derivative Test

Calculus 1 · Axiom Academy

LESSON The Second Derivative Test A flat tangent tells you where the extrema are; concavity tells you which kind . 1. A Flat Tangent Isn't Enough A critical point is an x = c where f'(c) = 0 (or f' doesn't exist). There the tangent line is horizontal, so the function momentarily stops climbing or falling. Watch the tangent below tilt up, then go perfectly flat at the marked point. A horizontal tangent — but is it a peak, a valley, or neither? 2. Concavity Is the Sign of f'' The second derivative is the slope of the slope — it measures how fast the tangent's tilt is changing. Watch a tangent sweep across each curve below: on the left the tilt keeps rising , on the right it keeps falling . The slope is increasing, so . The curve cups upward like a bowl. The slope is decreasing, so . The curve arches downward like a dome. 3. Reading the Sign at the Critical Point Zoom in on a critical point and the curve looks like a parabola — and f''(c) decides which way it opens. Watch the same flat point become a valley when , then flip to a hill when . If : f has a local minimum at c . If : f has a local maximum at c . If f''(c) = 0 : the test is inconclusive — fall back to the first-derivative sign test. 4. The Test in Action: f(x) = x^3 - 3x Differentiate once for the critical points, once more for the verdict. The animation slides along the real cubic and stops at each critical point to evaluate f'' . , then test each with f''(x) = 6x .

This is the written version of the interactive lesson above. See the full Calculus 1 course.