At x=1, the function f(x)={cases} x³-1 & 1

Correct Answer is : continuous and non-differentiable.

continuous and non-differentiable.

discontinuous and differentiable.

discontinuous and non-differentiable.

Correct Answer is : continuous and non-differentiable.

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KCET 2021 Math Solved Paper

Question : 37 of 60

Marks: +1, -0

x=1

f(x)=\begin{cases} x^3-1 & 1 < x < \infty \\ x-1 & -\infty < x \le 1 \end{cases}

Solution:

f(x)=\begin{cases} x^3-1 & 1 < x < \infty \\ x-1 & -\infty < x \le 1 \end{cases}

We have to check the continuity at

x=1

\text{RHL} \Rightarrow \lim\limits_{x\to 1^{1+}} f(x) = \lim\limits_{x\to 1^{1+}} (x^3-1) = 1-1 = 0

\text{LHL} \Rightarrow \lim\limits_{x\to 1^{1+}} f(x) = \lim\limits_{x\to 1^{1+}} (x-1) = 1-1 = 0

f(1)=1-1=0

Thus the function is continuous at

x=1

f(x)=\begin{cases} 3x^2 & 1 < x < \infty \\ 1 & -\infty < x \le 1 \end{cases}

Now, check the differentiability at

x=1

LHD at

x=1 \Rightarrow 1

RHD at

x=1 \Rightarrow 3(1)^2 = 3

As, LHD

\neq

RHD, function is not differentiable.

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