6. $f$ is concave down and $f(3)=-1$ and $f^{\prime}(3)=2$. a. What is the estimate for $f(3.2)$ using the local linear approximation for $f$ at $x=3$ ? b. Is it an underestimate or overestimate? Explain.

Step 1 ：The local linear approximation of a function \(f\) at a point \(x=a\) is given by the equation of the tangent line to the graph of \(f\) at \(x=a\). This is given by the formula \(L(x) = f(a) + f'(a)(x-a)\).

Step 2 ：In this case, we have \(a=3\), \(f(a) = f(3) = -1\), and \(f'(a) = f'(3) = 2\). So, the local linear approximation of \(f\) at \(x=3\) is \(L(x) = -1 + 2(x-3)\).

Step 3 ：We can use this to estimate the value of \(f(3.2)\) as follows: \(L(3.2) = -1 + 2(3.2-3) = -1 + 2(0.2) = -1 + 0.4 = -0.6\).

Step 4 ：So, the estimate for \(f(3.2)\) using the local linear approximation for \(f\) at \(x=3\) is \(\boxed{-0.6}\).

Step 5 ：Since \(f\) is concave down, the graph of \(f\) is below its tangent line. Therefore, the local linear approximation at \(x=3\) overestimates the value of \(f(3.2)\).