A function is twice differentiable for all real numbers
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A function f is twice differentiable for all real numbers. It is known that f′′(x)>0 for all x in the interval (−2,2). The point (0,4) lies on the graph of f, and the slope of the tangent line at x=0 is f′(0)=−3.
(a) Without performing any arithmetic calculations, determine whether using the tangent line at x=0 to approximate f(−0.3) will produce an underestimate or an overestimate of the actual value of f(−0.3). Explain your reasoning fully in terms of concavity.
(b) Using the same reasoning, classify the approximation of f(0.3) obtained from the tangent line at x=0 as an underestimate or an overestimate. Provide a clear justification.