Question

Let $f\left(x\right)=\{\begin{array}{c}ax,x<2\\ a{x}^2-bx+3,x\ge 2\end{array}$
If f(x) is differentiable for all x, then

• A. $a=\frac{3}{4}b=\frac{9}{4}$
• B. $a=1,b=2$
• C. $a=\frac{3}{2},b=\frac{9}{2}$
• D. $a=\frac{3}{4},b=\frac{11}{4}$

Answer 1

The correct option is A $a=\frac{3}{4}b=\frac{9}{4}$

f(x) is differentiable $\Rightarrow$ f(x) is continuous

f(x) is continuous at x = 2 $\Rightarrow$ 2a = 4a - 2b + 3 $\Rightarrow$ 2a - 2b + 3 = 0 . . . .(1)

f(x) is differentiable at x = 2 $\Rightarrow$ a = 4a - b $\Rightarrow$ 3a - b = 0 . . . .(2)

solving (1) and (2), we have $a=\frac{3}{4},b=\frac{9}{4}$