Let and be twice–differentiable on the open interval
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let f and g be twice–differentiable on the open interval (−1,5).
• The derivative f′(x) is strictly increasing on (−1,5) and satisfies f′(−1)=−3 and f′(5)=4.
• The derivative g′(x) behaves as follows:
– on (−1,1) it is strictly decreasing from 6 to 1;
– on (1,3) it is constant and equal to 1;
– on (3,5) it is strictly increasing from 1 to 5.
Assume no other critical behavior occurs for f′ or g′ on (−1,5).
(a) Determine the intervals on which the graph of f is concave up and the intervals on which it is concave down. Explain your reasoning.
(b) Determine the intervals on which the graph of g is concave up and the intervals on which it is concave down. Explain your reasoning.
(c) On the interval (0,4), decide which function, f or g, is "more concave upward." State your conclusion and provide a justification that compares the concavity of the two functions.
(d) Decide whether either function has a point of inflection at x=2. Justify your answer for each function.