Let where is a non–zero real parameter

MATHEMATICS · Subjective · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

f(x)=x^{3}-3a x^{2}+3a^{2}x

where

a

is a non–zero real parameter. (a) Show that

x=a

is a critical point of

f

. (b) Obtain

f''(a)

in terms of

a

and determine the nature of the critical point

x=a

for (i)

a>0

and (ii)

a<0

. (c) Find the local extreme value of

f

x=a

in each of the above cases. (d) For

a>0

, find any other critical point(s) of

f

, classify them, and state the intervals on which

f

is increasing or decreasing.