MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
Consider the polynomial function p(x)=x3−6x2+9x+2. The point (1,6) is on its graph.
(a) Compute p′′(x).
(b) Determine all open intervals containing x=1 on which the graph of p is concave up and all such intervals on which it is concave down. Show the algebraic steps leading to your conclusions.
(c) For each interval identified in part (b), state whether the tangent line to p at x=1 gives an underestimate or an overestimate of p(x) for x=1 in that interval. Provide a complete justification based on the concavity results from part (b).
(d) Using your work above, explain how the sign of p′′(x) governs the position of the graph of p relative to its tangent line at x=1 for values of x close to 1.