Mathematics

Problem

For each real number

k

, define

f_k(x)=x^{3}-3x+k

for

x

in the closed interval

[0,2]

. a. Compute

f_k(0)

and

f_k(2)

in terms of

k

. Simplify your expressions. b. Find all real values of

k

for which there exists at least one number

c

[0,2]

satisfying

f_k(c)=1

. Provide a complete derivation of the required set of

k

-values. c. For the specific choice

k=5

, verify whether a number

c

[0,2]

with

f_k(c)=1

must exist. Fully justify your reasoning without explicitly solving for

c