Let be a twice-differentiable polynomial whose first derivative is k'(x)=(x-2)^2(x+1)(x-5)
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let k be a twice-differentiable polynomial whose first derivative is
k′(x)=(x−2)2(x+1)(x−5).
Throughout this problem, fully justify each conclusion.
(a) Compute k′′(x) and determine all real numbers at which k′′(x)=0.
(b) Construct a complete sign chart for k′(x) and k′′(x) on the real line. Use the chart to list every open interval on which k is increasing and concave down.
(c) Verify that the intervals you found in part (b) are the only intervals on which k is simultaneously increasing and concave down.
(d) More generally, suppose p is any twice-differentiable polynomial whose first derivative has exactly four distinct real zeros. Prove that p can have at most three disjoint intervals on which it is simultaneously increasing and concave down.