Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Prove that the greatest integer function defined by
$\text{f(x)} = [\text{x} ], 0 < \text{x} < 3$
is not differentiable at x = 1 and x = 2.
✓

Answer

Given: $\text{f(x)} = [\text{x} ], 0 < \text{x} < 3$
$\text{R}\text{f}{'}(1) = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{\text{f(1 + h)}-\text{f}(1)}{\text{h}} = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{|1 + \text{h} |-1}{\text{h}}$$ = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{1-1}{\text{h}}= ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{0}{\text{h}}=0$
$\text{And}\ \text{L}\text{f}{'}(1) = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{\text{f(1 - h)}-\text{f}(1)}{-\text{h}} = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{|1 - \text{h}| - 1}{-\text{h}}$$ = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{|1-\text{h}|-1}{\text{h}}= ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{0-1}{-\text{h}}=\infty$
Since $\text{R }\text{f}{'}(1)\neq \text{L}{\text{f}}{'}(1)$
Therefore, f(x) = [x] is not differentiable at x =1.
Similarly f(x) = [x] is not differentiable at x = 2.

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 "3 Marks Question" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

In the matrix $\text{A}=\begin{bmatrix}\text{a}&1&\text{x}\\2&\sqrt{3}&\text{x}^2-\text{y}\\0&5&\frac{-2}{5}\end{bmatrix},$ write:
  1. The order of the matrix $A.$
  2. The number of elements.
  3. Write elements $a_{23}, a_{31}, a_{12}.$
→
Evaluate the following integrals:
$\int\frac{\text{x}^6+1}{\text{x}^2+1}\text{dx}$
→
Show that $\text{AB}\neq\text{BA}$ in the following cases:
$\text{A}=\begin{bmatrix}5&-1\\6&7\end{bmatrix}$ and $\text{B}=\begin{bmatrix}2&1\\3&4\end{bmatrix}$
→
Write a value of $\int\frac{1}{\text{x}(\log\text{x})^{\text{n}}}\text{ dx}$
→
Let $X$ be a random variable which assumes values $x_1, x_2, x_3, x_4$ such that $2 P\left(X=x_1\right)=3 P\left(X=x_2\right)=P\left(X=x_3\right)=5 P(X=$ $x_4$ ). Find the probability distribution of $X$.
→
Write the anti-derivative of $\Big(3\sqrt{\text{x}}+\frac{1}{\sqrt{\text{x}}}\Big)$
→
Without using the derivative, show that the function f(x) = |x| is
  1. Strictly increasing in $(0,\infty)$
  2. Strictly decreasing in $(-\infty,0)$
→
Find the coordinates of the foot of the perpendicular drawn from the origin.
$2x + 3y + 4z - 12 = 0$
→
Find the slope of the tangent to the curve $x = t^2 + 3t - 8, y = 2t^2 - 2t - 5$ at $t = 2.$
→
Given the matrices
$\text{A}=\begin{bmatrix}2&1&1\\3&-1&0\\0&2&4\end{bmatrix},\text{B}=\begin{bmatrix}9&7&-1\\3&5&4\\2&1&6\end{bmatrix}$ and $\text{C}=\begin{bmatrix}2&-4&3\\1&-1&0\\9&4&5\end{bmatrix}$ Verify that (A + B) + C = A + (B + C).
→