Consider the twice–differentiable function whose first derivative is given by…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Consider the twice–differentiable function f whose first derivative is given by
f′(x)=(x+4)(x−1)2(x−5).
Answer the following questions, showing all supporting analysis.
(a) Find all critical numbers of f and classify each as the x–coordinate of a local maximum, local minimum, or neither. Justify your classifications.
(b) Compute f′′(x). Determine the x–coordinates of all points of inflection and the open intervals on which f is concave up and concave down.
(c) Identify all open intervals on which f is increasing and concave down simultaneously. Provide a clear justification that references the signs of f′ and f′′. Express your answer using interval notation.
(d) Suppose f(x) represents the height (in centimeters) of a plant x days after it is planted. Interpret your answer from part (c) in the context of the plant’s growth.