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 is an overestimate of f(5.25) if the graph of f is concave down on the interval 5 < x < 5.25 (option C) because the function is curving downwards, making the tangent line lie above the actual function.

Step-by-step explanation:

The question asks which condition guarantees that the tangent line approximation of f(x) at x = 5.25 is an overestimate of f(5.25). To determine this, we must consider how the concavity of a twice-differentiable function affects the tangent line estimation.

If the graph of function f is concave down on the interval 5 < x < 5.25, it means that the actual function is curving downwards, while the tangent line at x = 5 would be a straight line. Since a concave down curve lies below the tangent line, the tangent line will provide an overestimate of the function values for values of x > 5, specifically at x = 5.25. Hence, the correct answer is C. The graph of the function f is concave down on the interval 5 < x < 5.25.