Question

Let $f\left(x\right)=\{\begin{array}{c}{x}^2,ifx\le x_0\\ ax+b,ifx>x_0\end{array}$lf $f$ is differentiable at $x_0$ then

• A. $a=x_0,b=-x_0$
• B. $a=2x_0,b=-x_0^2$
• C. $a=2x_0,b=x_0^2$
• D. $a=x_0,b=x_0^2$

Answer 1

Answer: (B)

$a=2x_0,b=-x_0^2$

$f\left(x\right)=\{\begin{array}{c}{x}^2,ifx\le x_0\\ ax+b,ifx>x_0\end{array}$

Since, $f$ is differentiable

Therefore, $f$ and $f^{\prime }$ are continuous function

For continuity of $f$

$LHL=RHL$

${x_0}^2=a{x}_0+b$ ....(1)

Now, $f^{\prime }\left(x\right)=\{\begin{array}{c}2x,ifx\le x_0\\ a,ifx>x_0\end{array}$

For continuity of $f^{\prime }$

$LHL=RHL$

$2x_0=a$ .....(2)

From (1) & (2)

$b=-(x_0{)}^2$