At the point x = 1,the given function f(x) = | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) We check the continuity at $x = 1$:$f(1) = 1 - 1 = 0$$LHL = \lim_{x 	o 1^-} f(x) = \lim_{h 	o 0} f(1 - h) = \lim_{h 	o 0} ((1 - h) - 1) = 0$$RH

Vedclass · Accepted Answer

Continuous and not differentiable

Vedclass · Answer

(B) We check the continuity at $x = 1$:$f(1) = 1 - 1 = 0$$LHL = \lim_{x 	o 1^-} f(x) = \lim_{h 	o 0} f(1 - h) = \lim_{h 	o 0} ((1 - h) - 1) = 0$$RHL = \lim_{x 	o 1^+} f(x) = \lim_{h 	o 0} f(1 + h) = \lim_{h 	o 0} ((1 + h)^3 - 1) = 0$Since $LHL = RHL = f(1)$,the function is continuous at $x = 1$.Now,we check the differentiability at $x = 1$:$Lf'(1) = \lim_{h 	o 0} \frac{f(1 - h) - f(1)}{-h} = \lim_{h 	o 0} \frac{(1 - h - 1) - 0}{-h} = \lim_{h 	o 0} \frac{-h}{-h} = 1$$Rf'(1) = \lim_{h 	o 0} \frac{f(1 + h) - f(1)}{h} = \lim_{h 	o 0} \frac{((1 + h)^3 - 1) - 0}{h} = \lim_{h 	o 0} \frac{1 + 3h + 3h^2 + h^3 - 1}{h} = \lim_{h 	o 0} (3 + 3h + h^2) = 3$Since $Lf'(1) 
eq Rf'(1)$,the function is not differentiable at $x = 1$.Thus,the function is continuous and not differentiable at $x = 1$.

At the point $x = 1$,the given function $f(x) = \begin{cases} x^3 - 1; & 1 < x < \infty \\ x - 1; & -\infty < x \le 1 \end{cases}$ is

Similar Questions

Let $S$ be the set of all points in $(-\pi, \pi)$ at which the function $f(x) = \min\{\sin x, \cos x\}$ is non-differentiable. Then $S$ is a subset of which of the following?

If $f(x) = a|\sin x| + be^{|x|} + c|x|^3$,where $a, b, c \in \mathbb{R}$,is differentiable at $x = 0$,then:

The set of all points where the function $f(x) = 2x|x|$ is differentiable is

Let $f(x) = x|x|$ and $g(x) = \sin x$.
Statement-$1$: $gof$ is differentiable at $x=0$ and its derivative is continuous at that point.
Statement-$2$: $gof$ is twice differentiable at $x=0$.

Assertion $(A)$: If $y = f(x) = (|x| - |x - 1|)^2$,then $\left(\frac{dy}{dx}\right)_{x=1} = 1$.
Reason $(R)$: If $\lim_{x \rightarrow a} \frac{f(x) - f(a)}{x - a}$ exists,then it is called the derivative of $f(x)$ at $x = a$.
Then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module