Question

Consider two functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to (1, \infty) \). Both functions are differentiable at a point \( c \). Which of the following functions is/are ALWAYS differentiable at \( c \)? The symbol \( \cdot \) denotes product and the symbol \( \circ \) denotes composition of functions.

• A. \( f \pm g \)
• B. \( f \cdot g \)
• C. \( \frac{f}{g} \)
• D. \( f \circ g + g \circ f \)

Hint:

The sum, product, and quotient of differentiable functions are always differentiable, as long as the denominator is non-zero. The composition of differentiable functions is also differentiable.

Answer 1

The Correct Option is A, B, C

We are given that both \( f \) and \( g \) are differentiable at point \( c \). Let's analyze each option:

Option (A): \( f \pm g \) is the sum or difference of two differentiable functions, which is always differentiable. Thus, Option (A) is correct.
Option (B): \( f \cdot g \) is the product of two differentiable functions, and the product of differentiable functions is always differentiable. So, Option (B) is correct.
Option (C): \( \frac{f}{g} \) is the quotient of two differentiable functions, and the quotient of differentiable functions is differentiable as long as the denominator \( g(x) \) is non-zero. Given that \( g \) maps to \( (1, \infty) \), \( g(x) \) is always positive, and thus \( \frac{f}{g} \) is differentiable. Therefore, Option (C) is correct.
Option (D): \( f \circ g + g \circ f \) represents compositions of differentiable functions. Composition of differentiable functions is differentiable, so Option (D) is also correct.

Thus, the correct answers are Option (A), Option (B), and Option (C).