Answer the following questions about monotonic behaviour on open and closed intervals.
a. (a) State the definition of an increasing function on (i) an open interval (a,b) and (ii) a closed interval [a,b].
b. (b) Let pk(x)=x3+kx, where k is real. Explain, without detailed computation, whether the condition pk′(x)≥0 for all x∈(a,b) is sufficient to conclude that pk is increasing on [a,b]. Justify your answer briefly.
c. (c) Using your reasoning in part (b), comment on any additional verifications needed when determining the set of k for which pk is increasing on the closed interval [−1,2].