P1: this problem is intended to help you deepen your understanding of optimality conditions for unconstrained continuous optimization. In developing the FONCs and SOSCs for unconstrained optimization, we made a number of assumptions, including that:

1. The objective function is continuous and real-valued over the domain it is defined, and

2. The objective function is also at least C1 smooth (first derivative is continuous) over this domain

Propose an objective function that has a local minimum, but that violates at least one of these assumptions in a way such that FONCs and SOSCs are not satisfied at the minimum. Discuss briefly why continuity and smoothness is required for the optimality conditions we derived to work for identifying local minima.

P1: This problem is intended to help you deepen your understanding of optimality conditions for unconstrained continuous optimization. In developing the FONCs and SOSCs for unconstrained optimization, we made a number of assumptions, including that: 1. The objective function is continuous and real-valued over the domain it is defined, and 2. The objective function is also at least C1 smooth (first derivative is continuous) over this domain Propose an objective function that has a local minimum, but that violates at least one of these assumptions in a way such that FONCs and SOSCs are not satisfied at the minimum. Discuss briefly why continuity and smoothness is required for the optimality conditions we derived to work for identifying local minima