GATE 2025 DA Q.51: solution and answer

Official answer: B, C, DSource: Official answer key, GATE 2025 DA (IIT Roorkee)

1. Since $f^{\prime \prime }(x)>0$ for all $x$, the derivative $f^{\prime }$ is strictly increasing, so $f$ is strictly convex.

2. (A) Not always. Take $f(x)=e^x$. Then $f^{\prime \prime }(x)=e^x>0$ everywhere, but $f^{\prime }(x)=e^x>0$ never vanishes, so $f$ has no stationary point and hence no local minimum. False.

3. (B) A strictly increasing function is one-to-one, so $f^{\prime }(x)=f^{\prime }(y)$ forces $x=y$. Hence two distinct points $x\ne y$ with $f^{\prime }(x)=f^{\prime }(y)=0$ cannot exist. True.

4. (D) $f$ is differentiable, so at any local minimum $c$ we need $f^{\prime }(c)=0$. By (B) there is at most one such point, so there is at most one local minimum. True.

5. (C) Suppose $f$ attained its global minimum value $m$ at two points $a\ne b$, so $f(a)=f(b)=m$. Strict convexity gives $f\left(\frac{a+b}{2}\right)<\frac{f(a)+f(b)}{2}=m$. This contradicts $m$ being the global minimum. So there is at most one global minimum point. True.

6. The statements that always hold are (B), (C) and (D).