Download App

Differentiation — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsDifferentiation3 Marks
Question
Determine whether the following function is differentiable at $x=3$ where,
$f(x)=x^2+2, \text { for } x \geq 3$
$=6 x-7, \text { for } x<3$
✓

Answer

$f(x)=x^2+2, x \geq 3$
$=6 x-7, x<3$
Differentiability at $x=3$
$
\begin{aligned}
\operatorname{Lf}^{\prime}(3) & =\lim _{h \rightarrow 0^{-}} \frac{f(3+h)-f(3)}{h} \\
& =\lim _{h \rightarrow 0^{-}} \frac{6(3+h)-7-\left(3^2+2\right)}{h} \\
& =\lim _{h \rightarrow 0^{-}} \frac{18+6 h-7-11}{h} \\
& =\lim _{h \rightarrow 0^{-}} \frac{6 h}{h}
\end{aligned}
$
$=\lim _{h \rightarrow 0^{-}} 6 \quad \ldots[\because h \rightarrow 0, \therefore h \neq 0]$
$=6$
$\operatorname{Rf}^{\prime}(3)=\lim _{h \rightarrow 0^{+}} \frac{f(3+h)-f(3)}{h}$
$=\lim _{h \rightarrow 0^{+}} \frac{(3+h)^2+2-\left(3^2+2\right)}{h}$
$=\lim _{h \rightarrow 0^{+}} \frac{h^2+6 h+9+2-11}{h}$
$=\lim _{h \rightarrow 0^{+}} \frac{h^2+6 h}{h}$
$=\lim _{h \rightarrow 0^{+}}(h+6) \ldots[\because h \rightarrow 0, \therefore h \neq 0]$
$=6$
$\text { Here, Lf }(3)=\text { Rf' }(3)$
$\therefore \text { fis differentiable at } x=3 .$

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 "Solve the Following Question.(3 Marks)" from Differentiation→Differentiation — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\omega$ is a complex cube root of unity, prove that $\left(1-\omega+\omega^2\right)^6+\left(1+\omega-\omega^2\right)^6=128$.
→
Evaluate the following:
$^{n+1}C_n$
→
Prove that following : $\cos 20^{\circ} \cos 40^{\circ} \cos 60^{\circ} \cos 80^{\circ}=\frac{1}{16}$
→
Show that the following are equivalence relations:

R in A is set of all books given by R = {(x, y) / x and y have same number of pages}

→
Find the equation of tangent to the circle $x^2+y^2-4 x+3 y+2=0$ at the point $(4,-2)$.
→
Write the following relations as sets of ordered pairs and find which of them are functions:
$\big\{(\text{x},\text{y})=\text{x}+\text{y}=3,\text{x},\text{y}\in\{0,1,2,3\}\big\}$
→
If $\sin\text{x}=\frac{4}{5}$ and $0<\text{x}<\frac{\pi}{2},$ find the value of sin 4x.
→
If a function f : R → R be defined by:
$\text{f(x)}=\begin{cases}3\text{x}-2,&\text{ x}<0\\1,&\text{x}=0\\4\text{x}+1,&\text{ x}>0\end{cases}$
Find: f(1), f(-1), f(0) and f(2)
→
Find the equation of the straight line which passes through the point (-3, 8) and cuts off positive intercepts on the coordinate axes whose sum is 7.
→
Show that for any sets A and B,$\text{A}=(\text{A}\cap\text{B})\cap(\text{A - B})$
→