Q. 20 - Calculus

Question

Suppose f is a piecewise-defined function, equal to g(x) if x < 2 and h(x) if x ≥ 2, where g and h are continuous and differentiable everywhere. If g'(2) = h'(2), is the function f necessarily differentiable at x = 2? Why or why not?

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{The} \mathrm{function} \mathrm{f} \mathrm{will} \mathrm{also} \mathrm{be} \mathrm{differentiable} \mathrm{at} \mathrm{x} = 2 \mathrm{as} \mathrm{the} \mathrm{left} \mathrm{hand} \\ \mathrm{derivative} \mathrm{and} \mathrm{right} \mathrm{hand} \mathrm{derivative} \mathrm{of} \mathrm{f} \mathrm{are} \mathrm{equal} \mathrm{at} \mathrm{x}=2. \end{gathered}$$

Step 1. Given information is:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\begin{array}{l}\mathrm{g}\left(\mathrm{x}\right), \mathrm{if} \mathrm{x}<2\\ \mathrm{h}\left(\mathrm{x}\right), \mathrm{if} \mathrm{x}\ge 2\end{array}\right. \\ \mathrm{and} \mathrm{g}\text{'}\left(2\right) = \mathrm{h}\text{'}\left(2\right) \end{gathered}$$

Step 2. Result

$$\begin{gathered} \mathrm{Since} \mathrm{g}\text{'}(2) = \mathrm{h}\text{'}(2) \mathrm{it} \mathrm{means} \mathrm{that} \mathrm{the} \mathrm{functions} \mathrm{g} \mathrm{and} \mathrm{h} \mathrm{are} \mathrm{differentiable} \mathrm{at} \mathrm{x}=2. \\ \mathrm{Therefore}, \mathrm{the} \mathrm{function} \mathrm{f} \mathrm{will} \mathrm{also} \mathrm{be} \mathrm{differentiable} \mathrm{at} \mathrm{x} = 2 \mathrm{as} \mathrm{the} \mathrm{left} \mathrm{hand} \\ \mathrm{derivative} \mathrm{and} \mathrm{right} \mathrm{hand} \mathrm{derivative} \mathrm{of} \mathrm{f} \mathrm{are} \mathrm{equal} \mathrm{at} \mathrm{x}=2. \end{gathered}$$