Let be the polynomial function

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

f

be the polynomial function

f(x) = x^5 - 5x^3 + 4x.

(a) Find

f'(x)

and determine all critical numbers of

f

. (b) Classify each critical number as giving a relative maximum, relative minimum, or neither, using an appropriate test based on

f'(x)

. (c) Compute

f''(x)

and determine the intervals of concave up and concave down behavior. (d) Identify every point of inflection of

f

. (e) Using the information from parts (a)–(d), describe the overall shape of the graph of

f

, including end behavior.