Mathematics

Problem

For each real number

k

, define

F_k(x)=x^{3}-6x+k

for

x

in the closed interval

[0,4]

. The family of equations

F_k(x)=0

is studied below. a. Express

F_k(0)

and

F_k(4)

in terms of

k

. Simplify your results. b. Determine the complete set of real values of

k

for which

F_k(0)

and

F_k(4)

have opposite signs. Explain why, for every

k

in this set, at least one solution of

F_k(x)=0

lies in

(0,4)

. c. Let

k=5

. Show that

F_5(x)=0

possesses at least two distinct real solutions in

[0,4]

by identifying two disjoint subintervals, each guaranteed to contain a solution. Provide clear justification for your claim.