The set of points at which the function

Question:

The set of points at which the function $f(x)=\frac{1}{\log |x|}$ is not differentiable, is_____________

Solution:

The given function is $f(x)=\frac{1}{\log |x|}$.

For $f(x)$ to be defined,

$x \neq 0$ and $\log |x| \neq 0$

$\Rightarrow x \neq 0$ and $|x| \neq 1$

$\Rightarrow x \neq 0$ and $x \neq \pm 1$

Thus, the function f(x) is not defined when x = −1, x = 0 and x = 1

We know that, the logarithmic function is differentiable at each point in its domain. Every constant function is differentiable at each x ∈ R. Also, the quotient of two differentiable functions is differentiable.

So, the function $f(x)=\frac{1}{\log |x|}$ is not differentiable at $x=-1, x=0$ and $x=1$.

Thus, the set of points at which the function $f(x)=\frac{1}{\log |x|}$ is not differentiable is $\{-1,0,1\}$

The set of points at which the function $f(x)=\frac{1}{\log |x|}$ is not differentiable, is $\{\underline{-1}, \underline{0}, 1\}$

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