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Question

$$\displaystyle f(x)  =  \left\{\begin{matrix}ax;& x < 2 \\ ax^{2}-bx+3; & x \geq 2\end{matrix}\right.$$ If $$f$$ is differentiable for all $$x$$ then


A
a=34,b=94
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B
a=32,b=92
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C
a=1,b=2
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D
a=34,b=92
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Solution

The correct option is A $$\displaystyle a=\frac {3}{4}, b=\frac {9}{4}$$
For a function to be differentiable it has to be continuous.

$$\Rightarrow f(2^-)=f(2^+)=f(2)\Rightarrow 2a=4a-2b+3$$

$$\Rightarrow 2a-2b+3=0 ........(i) $$

Now,

$$\displaystyle f'(x) =  \left\{\begin{matrix}a;& x < 2 \\ 2ax-b; & x \geq 2\end{matrix}\right.$$

$$\because$$ L.H.D $$=$$ R.H.D$$\Rightarrow a =4a-b\Rightarrow b = 3a ...(ii)$$ 

Solving (i) and (ii) we get, $$\displaystyle a=\frac{3}{4}, b=\frac{9}{4}$$

Mathematics

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