Question

15 mark for review the line tangent to the graph of the twice - differentiable function f at the point x = 3 is used to approximate the value of f(3.25). which of the following statements guarantees that the tangent line approximation at x = 3.25 is an underestimate of f(3.25)? a the function f is decreasing on the interval 3 ≤ x ≤ 3.25. b the function f is increasing on the interval 3 ≤ x ≤ 3.25. c the graph of the function f is concave down on the interval 3 ≤ x ≤ 3.25. d the graph of the function f is concave up on the interval 3 ≤ x ≤ 3.25.

Explanation:

When a function \(y = f(x)\) is concave - up on an interval \([a,b]\), the tangent line to the graph of the function at any point \(x = c\in[a,b]\) lies below the graph of the function on the interval \((a,b)\). Here, \(a = 3\), \(b=3.25\). If \(f\) is concave - up on the interval \(3\leq x\leq3.25\), then the tangent line approximation of \(f(3.25)\) (using the tangent line at \(x = 3\)) will be an underestimate of the actual value of \(f(3.25)\). When a function is increasing or decreasing, it doesn't guarantee that the tangent - line approximation is an underestimate. When a function is concave down, the tangent line lies above the graph and the tangent - line approximation is an overestimate.

Answer:

D. The graph of the function \(f\) is concave up on the interval \(3\leq x\leq3.25\)