Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY5 Marks
Question
Show that the function $\text{f(x)}\begin{cases}\text{x}^\text{m}\sin(\frac{1}{\text{x}}), &\text{x}\neq0 \\0 ,& \text{x}=0\end{cases}$
Differential at x = 0, if m > 1
✓

Answer

Let m = 2, then the function $\text{f(x)}=\begin{Bmatrix}\text{x}^2\sin(\frac{1}{\text{x}}) &\text{x}\neq0 \\0 & \text{x}=0 \end{Bmatrix}$
Differentiability at x = 0:
$\lim_\limits{\text{x}\rightarrow0}\frac{\text{f(x)}-\text{f(0)}}{\text{x}-0}=\lim_\limits{\text{x}\rightarrow0}\frac{\text{f(x)}}{\text{x}}=\lim_\limits{\text{x}\rightarrow0}\text{x}\sin\Big(\frac{1}{\text{x}}\Big)=0.$
$\big[\therefore\lim_\limits{\text{x}\rightarrow0}\text{x}\sin\Big(\frac{1}{\text{x}}\Big)\text{x}=0,$ as
$\begin{vmatrix}\text{x}\sin\frac{1}{\text{x}}-0\end{vmatrix}=\begin{vmatrix}\text{x}\sin\frac{1}{\text{x}} \end{vmatrix}=\begin{vmatrix}\text{x} \end{vmatrix}\begin{vmatrix}\sin\frac{1}{\text{x}} \end{vmatrix} \leq\begin{vmatrix}\text{x}\end{vmatrix}$
$\because\ |\sin\theta|\leq1\ \text{for all }\theta$
Hence, $\begin{vmatrix}\text{x}\sin\frac{1}{\text{x}} \end{vmatrix}<0\ \text{when}\ |\text{x-0}|<\epsilon|\text{x}-0|<\epsilon\big]$
Therefore, f'(x) = 0, which means f is differentiable at x = 0.
Hence the given function is differentiable at x = 0.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the direction cosines of the lines, connected by the relations: $l + m + n = 0$ and $\frac{2}{\text{m}}+\frac{2}{\text{n}}-\text{mn}=0$.
→
Find a point on the curve $y = x^3 - 3x$ where the tangent is parallel to the chord joining $(1, -2)$ and $(2, 2).$
→
Find the vector equation of the plane which is at a distance of $\frac{6}{\sqrt{29}}$ from the origin and its normal vector from the origin is $2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$ Also, find its cartesian form.
→
Find the value of k if the function: $f (x) = \begin{matrix} kx^{2}& x\geq 1 & \\ 4 & x< 1 & \\ \end{matrix} \text{is continuous at x = 1} $ 
→
Form the differential equation of the family of curve represented by $y^2 = (x - c)^3$
→
Evaluvate the following intregals:
$\int\frac{1}{\text{p}+\text{q}\tan\text{x}}\ \text{dx}$
→
The bag $A$ contains $8$ white and $7$ black balls while the bag $B$ contains $5$ white and $4$ black balls. One ball is randomly picked up from the bag $A$ and mixed up with the balls in bag $B$. Then a ball is randomly drawn out from it. Find the probability that ball drawn is white.
→
Using the method of integration, find the area of the region bounded by the lines
$\text{2x + y = 4, 3x - 2y = 6 and x -3y + 5 = 0}$
→
Evaluate the following integrals:$\int\frac{2\text{x}-3}{\text{x}^2+6\text{x}+13}\text{ dx}$
→
If $x^x + y^x = 1,$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}(\text{y}+\text{x}\log\text{y})}{\text{x}(\text{y}\log\text{x}+\text{x})}$
→