Question

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

Answer 1

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

For f(x) to be defined,

$x\ne 0$ and $\log \left|x\right|\ne 0$

$\Rightarrow x\ne 0$ and $\left|x\right|\ne 1$

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

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 $f\left(x\right)=\frac{1}{\log \left|x\right|}$ is not differentiable at x = −1, x = 0 and x = 1.

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


The set of points at which the function $f\left(x\right)=\frac{1}{\log \left|x\right|}$is not differentiable, is ___{−1, 0, 1}___.