Question

the locally linear approximation of the differentiable function f at x = 3 is used to approximate the value of f(3.2). the approximation at x = 3.2 is an overestimate of the corresponding function value at x = 3.2. which of the following could be the graph of f?

Explanation:

Step1: Recall linear - approximation property

The local linear approximation (tangent - line approximation) of a function \(y = f(x)\) at \(x=a\) is given by \(L(x)=f(a)+f^{\prime}(a)(x - a)\). If the local linear approximation at \(x = 3\) over - estimates \(f(3.2)\), then the function \(y = f(x)\) is concave down on the interval \((3,3.2)\).

Step2: Analyze concavity of each graph

• Graph A is a straight - line. For a straight - line, the linear approximation is exact (since the function is linear), so it cannot be the graph of \(f\) as we need an over - estimate.
• Graph B is concave down. For a concave - down function, the tangent line at a point \(x = a\) lies above the function for \(x\) values in a neighborhood of \(a\). If we take the tangent line at \(x = 3\) for a concave - down function, it will over - estimate the value of the function at \(x=3.2\).
• Graph C is concave up. For a concave - up function, the tangent line at a point \(x = a\) lies below the function for \(x\) values in a neighborhood of \(a\), so the tangent - line approximation at \(x = 3\) will under - estimate \(f(3.2)\).
• Graph D has a horizontal tangent at the point near \(x = 3\), and it is concave down to the left and concave up to the right. But in the interval \((3,3.2)\), we need a function that is concave down throughout this sub - interval for the tangent - line approximation at \(x = 3\) to over - estimate \(f(3.2)\), and this graph does not satisfy that condition.

Answer:

B.