Consider the function which maps a vector to its maximum entry, ↦ maxᵢ ᵢ. While this function is non-smooth, a common trick in machine learning is to use a smooth approximation, LogSumExp, defined as follows.
LSE : R" → R, LSE(x) = ln » R, LSE(P) = ln [ (i=1) Σ eˣᶦ]
One of the nice properties of this function is that it is convex, which can be proved by showing its Hessian matrix is positive semidefinite.
To that end, compute its gradient and Hessian.

Tom is the deli manager at a grocery store. he needs to schedule employee to staff the deli department for no more that 260 person-hours per week. tom has one part-time employee who works 20 person-hours per week. each full-time employee works 40 person-hours per week. write and inequality to determine n, the number of full-time employees tom may schedule, so that his employees work on more than 260 person-hours per week. graph the solution set to this inequality.