Download App

Differentiation — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsDifferentiation5 Marks
Question
Determine the values of $p$ and $q$ that make the function $f(x)$ differentiable on $R$ where
$f(x) = px^3,$ for $x < 2$
$= x^2 + q,$ for $x \geq 2$
✓

Answer

$\therefore f ( x )$ is differentiable at $x =2$.
$\therefore f ( x )$ is continuous at $x =2$.
Continuity at $x=2$ :
$f(x)$ is continuous at $x=2$.
$\therefore \quad \lim _{x \rightarrow 2^{-}} f (x)=\lim _{x \rightarrow 2^{+}} f (x)$
$\therefore \quad \lim _{x \rightarrow 2^{-}} p x^3=\lim _{x \rightarrow 2^{+}}\left(x^2+ q \right)$
$\therefore \quad 8 p =4+ q \text { }$
$\therefore \quad 8 p - q =4$
$\text { Differentiability at } x= 2 \text { : }$
$L f^{\prime}(2)=\lim _{h \rightarrow 0^{-}} \frac{f(2+h)-f(2)}{h}$
$=\lim _{h \rightarrow 0^{-}} \frac{p(2+h)^3-\left(2^2+q\right)}{h}$
$=\lim _{b \rightarrow 0^{-}}\left(\frac{ ph ^3+6 ph ^2+12 hp +8 p -(4+ q )}{ h }\right)$
$=\lim _{h \rightarrow 0^{-}} \frac{h\left( ph ^2+6 ph +12 p \right)}{ h }$
$\ldots[\because 8 p =4+q]$
$=\lim _{h \rightarrow 0^{-}}\left( ph ^2+6 ph +12 p \right)$
$\ldots[\because h \rightarrow 0, \therefore h \neq 0]$
$=12 p$
$R f^{\prime}(2)=\lim _{h \rightarrow 0^{+}} \frac{f(2+h)-f(2)}{h}$
$=\lim _{h \rightarrow 0^{+}} \frac{(2+h)^2+q-\left(2^2+q\right)}{h}$
$=\lim _{h \rightarrow 0^{+}} \frac{h^2+4 h+4+q-(4+q)}{h}$
$=\lim _{h \rightarrow 0}\left(\frac{h^2+4 h}{h}\right)$
$=\lim _{h \rightarrow 0}(h+4) \quad \ldots[\because h \rightarrow 0, \therefore h \neq 0]$
$=4$
$f(x)$ is differentiable at $x=2$.
$\therefore Lf ^{\prime}(2)= Rf ^{\prime}(2)$
$\therefore 12 p =4$
$\therefore p =\frac{1}{3}$
Substituting $p=\frac{1}{3}$ in (i), we get
$8\left(\frac{1}{3}-q=4\right.$
$\therefore q=\frac{8}{3}-4=\frac{-4}{3}$

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.(5 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

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{{\pi}}}\frac{1-\sin\frac{\text{x}}{2}}{\cos\frac{\text{x}}2\big(\cos\frac{\text{x}}{2}-\sin\frac{\text{x}}{4}\big)}$
→
A person observes the angle of elevation of the peak of a hill from a station to be $\alpha.$ He walks c metres along a slope inclined at an angle $\beta$ and finds the angle of elevation of the peak of the hill to be $\gamma.$ Show that the height of the peak above the ground is $\frac{\text{c}\sin\alpha\sin(\gamma-\beta)}{(\sin\gamma-\alpha)}.$
→
Find the modulus and argument of the complex number $\frac{1+2 i}{1-3 i}$.
→
If $\frac{1 \times 3+2 \times 5+3 \times 7+\ldots \text { upto } n \text { terms }}{1^3+2^3+3^3+\ldots \text { upto } n \text { terms }}=\frac{5}{9}$, find the value of $n$.
→
Represent to solution set of each of the following inequations graphically in two dimensional plane:
$\text{y}>2\text{x}-8$
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow5}\frac{\text{x}-5}{\sqrt{6\text{x}-5}-\sqrt{4\text{x}+5}}$
→
Find the sum of the following series to n terms:
$1.2.4 + 2.3.7 +3.4.10 + ...$
→
Sum the following series to n terms:
$4 + 6 + 9 + 13 + 18 + .....$
→
If a circle passes through the point (0, 0),(a, 0),(0, b) then find the coordinates of its centre.
→
Find the equation of the hyperbola whoseFocus is at $(5, 2),$ vertex at $(4, 2)$ and center at $(3, 2)$
→