MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let f(x)=x3−6x2+9x+5 be defined for −1≤x≤5.
(a) Calculate f′(x) and determine all values of x in the interval [−1,5] at which f′(x)=0 or is undefined.
(b) Construct a sign chart for f′(x) on [−1,5]. Clearly indicate the sign of f′(x) on each sub-interval you identify.
(c) Using only the sign information from part (b) together with the endpoints, determine and justify the x-coordinate(s) of the absolute maximum and the absolute minimum of f on the interval [−1,5].
(d) Evaluate f at each point you identified in part (c) and state the absolute maximum value and the absolute minimum value of f on [−1,5]. Express your answers as ordered pairs (x,f(x)).