For a real parameter

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

Problem

m

, consider the family of quartic functions

q_m(x)=x^{4}-2m x^{2}+m^{2}

defined on the closed interval

[-3,3]

. (a) Find

q_m'(x)

and solve

q_m'(x)=0

to express all critical points of

q_m

in terms of

m

. Indicate which of these points lie in

[-3,3]

for various values of

m

. (b) Classify, in terms of

m

, the different cases for the number of distinct critical points

q_m

possesses in

[-3,3]

. (c) For each case identified in part (b), compare the values of

q_m

at the relevant critical points and at the endpoints

x=-3

and

x=3

. Derive a complete description of the global minimum value of

q_m

[-3,3]

as a function of

m

, and specify the

x

–location(s) where it occurs. Provide rigorous justification for every inequality used.