Скачать или смотреть Calculus 1 — 18.3: Concavity and Inflection Points

Can a function decrease yet still curve upward like a cup? The second derivative reveals the shape of a curve — its concavity — independent of whether the function is rising or falling. This lesson walks through the complete procedure for finding inflection points, from building sign charts to verifying that concavity actually changes.

Key concepts covered:
• First derivative measures direction; second derivative measures shape (concavity)
• Concave up (f'' positive) means slopes are increasing — cup shape
• Concave down (f'' negative) means slopes are decreasing — arch shape
• Possible inflection points (PIPs) occur where f'' equals zero or is undefined
• The 7-step sign chart procedure for classifying concavity intervals
• Polynomial example: f''(x) = 12x² − 24x with inflection points at (0, 0) and (2, 12)
• Cube root example: g(x) = (x − 1)^(1/3) where f'' is undefined at x = 1
• Counter-example: f(x) = x⁴ where f'' = 0 at x = 0 but no sign change occurs — not an inflection point
• Why f'' equaling zero is necessary but not sufficient for an inflection point
• Connection to the first derivative test: same logic, one level deeper

ORIGINAL SOURCE
This video is based on standard calculus curriculum covering the Second Derivative Test for concavity and inflection points.