GATE 2024 AG – Question 21
A continuously differentiable function y=f(x) satisfies which listed conditions at a point of inflection? Select all that apply.
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Correct answer: (B) $y^{\prime\prime}=0$; (D) $y^{\prime\prime\prime}\ne0$
Explanation
The official selections **B, D** describe the standard simple-inflection test: $f^{\prime\prime}(x_0)=0$ and $f^{\prime\prime\prime}(x_0)\ne0$ imply a concavity change under sufficient smoothness. These are not necessary conditions for every inflection under only the printed continuous-differentiability assumption. For example,$f=x^5$ has an inflection at 0 with third derivative0; some C¹ examples have undefined second derivative. The keyed sufficient-test interpretation is stated explicitly.