taiwan-gsat 2025 Q13

taiwan-gsat · Other · ast__math-b 3 marks Stationary points and optimisation Find concavity, inflection points, or second derivative properties

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$.
Regarding the statements about $f ^ { \prime \prime } ( - 3 )$ and $f ^ { \prime \prime } ( 1 )$, select the correct option. (Single choice)
(1) $f ^ { \prime \prime } ( - 3 ) = f ^ { \prime \prime } ( 1 ) = 0$
(2) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) > 0$
(3) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) < 0$
(4) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) > 0$
(5) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) < 0$

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$.

Regarding the statements about $f ^ { \prime \prime } ( - 3 )$ and $f ^ { \prime \prime } ( 1 )$, select the correct option. (Single choice)\\
(1) $f ^ { \prime \prime } ( - 3 ) = f ^ { \prime \prime } ( 1 ) = 0$\\
(2) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) > 0$\\
(3) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) < 0$\\
(4) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) > 0$\\
(5) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) < 0$

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