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.
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.
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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.
Updated On: Apr 4, 2025
- \( f \pm g \)
- \( f \cdot g \)
- \( \frac{f}{g} \)
- \( f \circ g + g \circ f \)
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The Correct Option is A, B, C
Solution and Explanation
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).
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).
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