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

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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