Question

The line tangent to the graph of the twice-differentiable function f at the point x = 5 is used to approximate the value of f(5.25). Which of the following statements guarantees that the tangent line approximation at x = 5.25 is an overestimate of f(5.25)?

A The function f is decreasing on the interval 5 ≤ x ≤ 5.25.
B The function f is increasing on the interval 5 ≤ x ≤ 5.25.
C The graph of the function f is concave down on the interval 5 ≤ x ≤ 5.25.
D The graph of the function f is concave up on the interval 5 ≤ x ≤ 5.25.

Answer 1

The tangent line approximation at x = 5.25 will be an overestimate if the graph of function f is concave down between x = 5 and x = 5.25 (Option C). This is because the slope decreases, making the tangent line sit above the curve past the point of tangency.

Step-by-step explanation:

The line tangent to the graph of a twice-differentiable function f at a certain point can be used to approximate the value of the function near that point. When determining whether the tangent line approximation is an overestimate or an underestimate of f(5.25), we must consider the concavity of the function and the direction in which the function is moving.

If the graph of the function f is concave down on the interval between 5 and 5.25 (Option C), it means that the slope of the function is decreasing as x increases. In this scenario, the line tangent at x = 5 will lie above the graph for values of x greater than 5, since the actual graph is curving downwards away from the tangent line. Thus, the tangent line will provide an overestimate of f(5.25). On the other hand, if the graph is concave up or the function is increasing, then the tangent line might underestimate or overestimate the function value depending on the exact behavior of the function in the interval. Therefore, the correct option that guarantees an overestimate is Option C.