MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A twice–differentiable function g is defined on −4≤x≤7. A textual description of its graph is provided.
Graph of g:
• The curve passes through (−4,5) with a horizontal tangent.
• On (−4,0) the curve is concave down and decreasing; at x=0 the curve has a point of inflection with g(0)=2.
• On (0,3) the curve is concave up and decreasing until x=3, where a local minimum occurs at (3,1).
• From x=3 to x=5 the curve is increasing and concave up.
• At x=5 the curve has another point of inflection, after which on (5,7] the curve is increasing but concave down, ending at (7,6).
(a) Using only the concavity information, construct a detailed sign chart for g′′ on (−4,7). State explicitly on which open intervals g′′(x)>0 and on which g′′(x)<0.
(b) Identify every x–value in (−4,7) where g′′(x)=0. For each, provide the corresponding feature of the graph of g that supports your conclusion.
(c) Sketch a possible graph of g′′ on −4≤x≤7, labeling intercepts and indicating whether each point where g′′(x)=0 is a simple zero or a change in sign.
(d) Explain how your sketch in part (c) is consistent with the behavior of the slopes of tangents to g described in the prompt. Your explanation should reference the concepts of slope of g′ and concavity in a coherent narrative.