For the family of functions F_k(x)=x^ 3 -3k x+7

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

Problem

For the family of functions

F_k(x)=x^{3}-3k x+7,

k

being a real parameter, consider the behaviour of stationary points. (i) Determine the coordinates of all stationary points in terms of

k

. (ii) Find the set of all real values of

k

for which the function possesses both a local maximum and a local minimum. (iii) For those values of

k

obtained in part (ii), identify which stationary point corresponds to the maximum and which to the minimum.