Consider the function defined by for all real .

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

Problem

Consider the function

f

defined by

f(x)=x^4-4x^3+10

for all real

x

. (a) Calculate all critical numbers of

f

and show your work clearly. (b) For each critical number found in part (a), determine whether

f

has a local maximum, local minimum, or neither at that point. Support each conclusion with analytic evidence. (c) Verify that

f

has exactly one local extremum on

\mathbb{R}

and state whether that extremum is absolute on

\mathbb{R}

. Justify your answer by connecting properties of

f

to your results from parts (a) and (b).