Concavity & Inflection Points

AP Calculus · Axiom Academy

Lesson Concavity &amp, 'Apple Color Emoji', 'Segoe UI Emoji', 'Noto Color Emoji'; Inflection Points Understanding how the second derivative reveals the shape of a curve Concavity: The Shape of a Curve We already know f'(x) tells us whether a function is increasing or decreasing. The second derivative f''(x) tells us how the function curves. f is concave up on I if f''(x) > 0 for all x in I (the curve bends upward, like a cup). f is concave down on I if f''(x) < 0 for all x in I (the curve bends downward, like a cap). Think of it this way: concave up means the slope f'(x) is increasing , and concave down means the slope is decreasing . A point (c, f(c)) is an inflection point if f changes concavity at x = c . That is, f''(x) changes sign at x = c . Find where f''(x) = 0 or f''(x) is undefined — these are candidates . Verify that f''(x) actually changes sign at each candidate. Find the intervals of concavity and any inflection points for f(x) = x^3 - 6x^2 + 9x + 1 . Step 1: f'(x) = 3x^2 - 12x + 9 x > 2 : f''(3) = 6(3) - 12 = 6 > 0 concave up Since f'' changes sign at x = 2 , the point (2, f(2)) = (2, 3) is an inflection point . f''(x) > 0 concave up (slope increasing) f''(x) < 0 concave down (slope decreasing) Inflection points occur where f''(x) changes sign Always verify sign change — f''(c) = 0 alone is not sufficient

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