Let $f_n(x) + f_n(y) = \frac{x^n + y^n}{x^n y^n}$ for all $x, y \in \mathbb{R} - \{0\}$ where $n \in \mathbb{N}$. Let $g(x) = \max\left\{f_2(x), f_3(x)\right\}$ for all $x \in \mathbb{R} - \{0\}$. The number of values of $x$ for which $g(x)$ is non-differentiable ($x \in \mathbb{R} - \{0\}$):

Limits, Continuity & Differentiability

Differentiability Analysis

Grade 12

Question:

Let $f_n(x) + f_n(y) = \frac{x^n + y^n}{x^n y^n}$ for all $x, y \in \mathbb{R} - \{0\}$ where $n \in \mathbb{N}$.Let $g(x) = \max\left\{f_2(x), f_3(x)\right\}$ for all $x \in \mathbb{R} - \{0\}$.The number of values of $x$ for which $g(x)$ is non-differentiable ($x \in \mathbb{R} - \{0\}$):

(a) 3

(b) 4

(c) 5

(d) 1

Step-by-Step Solution

Key Concept: Determine $f_n(x) = x^{-n} + x^n$. Find where $f_2(x) = f_3(x)$, i.e., where $x^{-2} + x^2 = x^{-3} + x^3$. The function $g(x)$ is non-differentiable at these points and where individual functions have non-differentiable points.

Answer: (a)

Correct Answer: A

Question:

Step-by-Step Solution

Master Limits, Continuity & Differentiability with Mathbee