Gate DA 2025 | Question - 24 | Calculus and Optimization

Given:

• $f: \mathbb{R} \rightarrow \mathbb{R}$

• $g: \mathbb{R} \rightarrow (1, \infty)$

• Both $f$ and $g$ are differentiable at a point $c \in \mathbb{R}$.

• We are to find which of the following functions is/are always differentiable at $c$.

Option (A): $f \pm g$

• The sum and difference of two differentiable functions are always differentiable.
  ✅ Correct

Option (B): $f \cdot g$

• The product of two differentiable functions is always differentiable.
  ✅ Correct

Option (C): $\frac{f}{g}$

• Division is differentiable only if the denominator is non-zero at c.

• Given $g(x) > 1 \Rightarrow g(c) > 1 \Rightarrow g(c) \ne 0$, so division is safe.
  ✅ Correct

Option (D): $f \circ g + g \circ f$

We analyze both compositions:

• $f \circ g$:
  $g(x) \in (1, \infty) \subset \mathbb{R} \Rightarrow g(x)$ is in domain of $f$
  So $f(g(x))$ is well-defined.
  $g$ is differentiable at $c$, and $f$ is differentiable at $g(c)$, hence $f \circ g$ is differentiable at $c$.

• $g \circ f$:
  $f(x) \in \mathbb{R} \Rightarrow f(x)$ might not lie in the domain of $g$ (which is $(1, \infty)$)
  So $g(f(x))$ may not be defined at all.

   

   

❌ Not always differentiable