Subtopic: Maxima and minima of function with several independent variables
Question Statement:
A continuously differentiable function y = f(x) satisfies one or more of the following conditions at its point of inflection:
Options:
(A) d²y/dx² ≠ 0
(B) d²y/dx² = 0
(C) d³y/dx³ = 0
(D) d³y/dx³ ≠ 0
Official Answer Key:
B, D
Step-by-Step Solution & Explanation
A point of inflection is where the curve changes its sense of concavity (concave up ↔ concave down).
(B) d²y/dx² = 0 — the second derivative measures concavity, so it must vanish where concavity changes sign. This is the necessary condition. ✔
(D) d³y/dx³ ≠ 0 — vanishing of y″ alone is not sufficient (e.g. y = x⁴ at x = 0 has y″ = 0 but is a minimum, not an inflection). A non-zero third derivative guarantees that y″ actually changes sign rather than merely touching zero. This is the sufficient condition. ✔
contradicts (B); (C) would leave the test inconclusive.
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