The function f(x)= x^2 a ,&0 <1 ,&1 <2 2b^2-4b x^2 ,&2 < is continuous for 0 < , then the most suitable values of a and b are: 1. a=1, b=-1 2. a=-1, b=1+2 3. a=-1, b=1 4. None os these.

3. a = -1, b = 1 Solution: Given, f(x) is continuous for 0 < . This means that f(x) is continuous for x=1,2. Now, If(x) is continuons at x = 1, then _ x 1^- f(x)=f(1) _ h 0 f(1-h)=a (1-h)^2 a =a 1 a =a ^2=1 = 1 If f(x) is continuous at x= 2 , then _ x 2^- f(x)=f(2) _ h 0 f(2-h)= 2b^2-4b 2 _ h 0 a=b^2-2b = b^2-2b ^2-2b - a=0 For a = -1, We have b^2-2b+1=0 (b-1)^2=0 =1 Thus, a = -1 and b=1

The function f(x)= x^2 a ,&0 <1 ,&1 <2 2b^2-4b x^2 ,&2 < is conti…

Download App

Question

The function $\text{f(x)}=\begin{cases}\frac{\text{x}^2}{\text{a}},&0\leq\text{x}<1\\\text{a},&1\leq\text{x}<\sqrt{2}\\\frac{2\text{b}^2-4\text{b}}{\text{x}^2},&\sqrt{2}\leq\text{x}<\infty\end{cases}$ is continuous for $0\leq\text{x}<\infty,$ then the most suitable values of a and b are:

$\text{a}=1,\text{ b}=-1$
$\text{a}=-1,\text{ b}=1+\sqrt{2}$
$\text{a}=-1,\text{ b}=1$
$\text{None os these}.$

✓

Answer

a = -1, b = 1

Solution:

Given, f(x) is continuous for $0\leq\text{x}<\infty.$

This means that f(x) is continuous for $\text{x}=1,\sqrt{2.}$

Now,

If(x) is continuons at x = 1, then

$\lim\limits_{\text{x}\rightarrow1^-}\text{f(x)}=\text{f}(1)$

$\Rightarrow\lim\limits_{\text{h}\rightarrow0}\text{f}(1-\text{h})=\text{a}$

$\Rightarrow\frac{(1-\text{h})^2}{\text{a}}=\text{a}$

$\Rightarrow\frac{1}{\text{a}}=\text{a}$

$\Rightarrow\text{a}^2=1$

$\Rightarrow\text{a}=\pm1$

If f(x) is continuous at $\text{x}={\sqrt{2}},$ then

$\lim\limits_{\text{x}\rightarrow\sqrt{2}^-}\text{f(x)}=\text{f}(\sqrt{2})$

$\Rightarrow\lim\limits_{\text{h}\rightarrow0}\text{f}(\sqrt{2}-\text{h})=\frac{2\text{b}^2-4\text{b}}{2}$

$\Rightarrow\lim\limits_{\text{h}\rightarrow0}\text{a=b}^2-2\text{b}$

$\Rightarrow\text{a = b}^2-2\text{b}$

$\Rightarrow\text{b}^2-2\text{b - a}=0$

$\therefore$ For a = -1, We have

$\text{b}^2-2\text{b}+1=0$

$\Rightarrow(\text{b}-1)^2=0$

$\Rightarrow\text{b}=1$

Thus, a = -1 and b=1

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 "M.C.Q (1 Marks)" 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→

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions