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).

Question

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 function f is concave down on the interval 3 ≤ x ≤ 3.25

D. The function f is concave up the interval 3 ≤ x ≤ 3.25

Answer 1

Solution:

The tangent line approximation is obtained by determining the slope of the tangent line or the derivative of the function f at 3.25. The derivative is the limit of the function f as x → 3.25 but without actually getting to it. In Calculus terms:

y = f (x) , and y' = f '(x) , and f '( 3.25) is the slope of tangent line at point x = 3.25

The value of the function f (3.25) is determined by substituting x = 3.25 in the original f (x) function

for the value of the tangent line, or the limit of f(x) as x → 3.25 to be an underestimate or less than the value of f ( 3.25) , the function must be increasing over the interval between 3 and 3.25. The concavity is the value of the 2nd derivative or twice differentiable function at point x = 3.25, and the function is concave up when that value is positive, and concave down when the value is negative. For the tangent line approximation to be under the actual value is valid when the function is increasing and concave up .This eliminates options a and c and leaves b and d. The increasing function does not guarantee an underestimation of the tangent line value because the line could be above the curve. However with the function concave up the tangent line value is always below the function value as x increases say from 3 to 3.25 , and therefore the answer is d

Answer 2