A function f is defined by
f(x)={2−x,(x−2)2+k,x<2,x≥2,
where k is a real constant. Consider the interval [−1,5].
(a) Determine the value of k that makes f continuous at x=2.
(b) With this value of k, compute f(−1) and f(5).
(c) Justify that the equation f(x)=4 has at least one real solution on [−1,5]. State an open sub-interval of [−1,5] that must contain such a solution.