A continuous function has a second derivative whose graph consists of three line…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A continuous function f has a second derivative f′′ whose graph consists of three line segments joining the points (−6,0), (−2,4), (1,−3), and (5,0). In addition, f′(−6)=2 and f(−6)=7.
(For clarity: from x=−6 to x=−2, f′′ increases linearly; from x=−2 to x=1, f′′ decreases linearly; from x=1 to x=5, f′′ increases linearly.)
(a) Determine the interval(s) on which the graph of f is concave up and the interval(s) on which it is concave down on −6<x<5. Justify your answer.
(b) Find the x–coordinate(s) of all inflection point(s) of f on the interval −6<x<5 and explain why each qualifies.
(c) Obtain an explicit piecewise formula for f′(x) on each of the intervals [−6,−2], [−2,1], and [1,5]. Express each linear expression in simplest form.
(d) Using the result from part (c), compute f(5). Show all steps, including any intermediate calculation of f′(1) or f(1) needed to preserve continuity.
(e) On the axes below, sketch a possible graph of y=f(x) on the interval −6≤x≤5. Label the inflection point(s) and indicate concave–up and concave–down regions on your sketch.