MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
Three smooth curves R, S, and T are drawn on the same set of coordinate axes. Their observed features are listed below.
• Curve S has horizontal tangent lines at x=−2 and x=3. At x=−2 the graph changes from increasing to decreasing, and at x=3 it changes from decreasing to increasing.
• Curve R intersects the x-axis at x=−5, x=−2, x=3, and x=6. It lies above the x-axis on (−5,−2)∪(3,6) and below the x-axis on (−2,3).
• Curve T is positive on (−2,3) and negative on (−5,−2)∪(3,6). It attains a local minimum at x=0 where its tangent line is horizontal.
Exactly one of these curves is the graph of a differentiable function f, one is the graph of f′, and one is the graph of f′′.
(1) Part 1: Identify which curve represents f, which represents f′, and which represents f′′. Provide a clear justification that refers explicitly to the listed features of each curve.
(2) Part 2: For the curve you identified as f, state the x-coordinate(s) of all relative extrema and classify each as a relative maximum or relative minimum. Justify your classifications.
(3) Part 3: For the curve you identified as f, determine the open interval(s) on which the function is concave downward. Explain how you used information from the curve you designated as f′′.