Your mark on the Calculus exam in percent, M(t), depends on how many hours you study, t.

M(t) = 5.2t(3kt) + 40

a) If after one hour of studying your mark would be 45%, determine the value of k.

b) What is your maximum mark (local max, no need to use the optimization algorithm), and how long do you need to study to achieve that mark?

c) Relating back to the context of the question, why does it make sense for M(t) to have a local maximum (as opposed to having either a local minimum or no local extrema)?