Determine \(L(x)\text{,}\) the local linear approximation to \(f(x)\) at \(x = 3\text{.}\) Explain why this function is also the local linear approximation to \(g(x)\) at \(x = 3\text{.}\)
On the axes provided in Figure 1.8.5, plot \(f(x)\text{,}\)\(g(x)\text{,}\) and \(L(x)\) near \(x = 3\text{.}\) In addition, compute \(f''(3)\text{,}\)\(g''(3)\text{,}\) and \(L''(3)\text{.}\) How do the second derivative values and the graph explain why \(L(x)\) is a better approximation for one of the functions than it is for the other?
