Question

Let

\(f\left( x \right) = \left\{ \begin{aligned}{x^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{if}}\,\,x \le 2\\mx + b\,\,\,\,\,\,\,\,\,\,{\rm{if}}\,\,x > 2\end{aligned} \right.\)

Find the values of \(m\) and \(b\) that make \(f\) differentiable everywhere

Short answer

The required values are \(m = 4\) and \(b = - 4\).

Step-by-step solution

Differential Functions

The differential functions are those functions for which the necessary and sufficient condition is function being continuous at every point within interval and derivatives must exist too.

Evaluating function for each interval.

Thegiven equation is:

\(f\left( x \right) = \left\{ \begin{aligned}{x^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{if}}\,\,x \le 2\\mx + b\,\,\,\,\,\,\,\,\,\,{\rm{if}}\,\,x > 2\end{aligned} \right.\)

Since,\(f\)is already seen differentiable at for all\(x < 2{\rm{ and }}x > 2\).

Now, for\(x < 2\), we have:

\(\begin{aligned}f'\left( x \right) &= 2x\\f'\left( x \right)\left| {_{x = 2}} \right. &= 2\left( 2 \right)\\ &= 4\end{aligned}\)

Similarly, for\(x > 2\), we have:

\(\begin{aligned}f'\left( x \right) &= m\\f{'_ + }\left( x \right)\left| {_{x = 2}} \right. &= m\end{aligned}\)

Checking differentiability for \(x = 2\)

Now,for function being differentiable at \(x = 2\), we have:

\(\begin{aligned}4 = f_ - '\left( 2 \right) &= f_ + '\left( 2 \right) = m\\ \Rightarrow f\left( x \right) &= mx + b\end{aligned}\)

And, for function being continuous at\(x = 2\), we have:

\(\begin{aligned}\;f\left( 2 \right) &= \mathop {\lim }\limits_{x \to {2^ + }} f\left( x \right)\\4 &= \mathop {\lim }\limits_{x \to {2^ + }} 4x + b\\4 &= 8 + b\\b &= - 4\end{aligned}\)

So, we have:

\(\begin{aligned}m =& 4\\b =& - 4\end{aligned}\)

Hence, the required values are \(m = 4\) and \(b = - 4\).