MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let h be a twice–differentiable function defined on −6≤x≤6. Its graph possesses the following features:
• On −6<x<−3 the graph is a straight line with slope 2.
• At x=−3 the graph has a sharp corner.
• Between x=−3 and x=1 the graph is the arc of the parabola y=−(x+1)2+4.
• At x=1 the graph transitions smoothly to a cubic segment that is concave down and decreasing on 1<x<4.
• At x=4 the graph has a horizontal tangent and changes to concave up on 4<x<6.
Analyze h′′ as follows.
(a) Construct a complete sign chart for h′′ on the interval −6<x<6. Clearly indicate intervals of positivity, negativity, zeros, and non–existence.
(b) Identify and classify every x–value in −6≤x≤6 at which h′′(x) does not exist. Provide a justification based on the given features of h.
(c) Sketch a possible graph of h′′ that is consistent with your sign chart from part (a) and the discontinuities from part (b). Label all intercepts, discontinuities, and important points.
(d) Explain how the concavity information embodied in your graph of h′′ is reflected in the original description of h. Cite specific intervals or points in your discussion.