Let be a real parameter and consider f(x)=x^ 4 -4k x^ 3 +6k^ 2 x^ 2 .

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

Problem

Let

k

be a real parameter and consider

f(x)=x^{4}-4k x^{3}+6k^{2}x^{2}.

(a) Find all critical points of

f

in terms of

k

. (b) Compute the corresponding values of

f

at these critical points. (c) Determine all real values of

k

for which the two distinct turning points of the curve have equal ordinate, that is,

f(c_{1})=f(c_{2})

where

c_{1}\ne c_{2}

are the critical points.