a-) Is the following statements TRUE or FALSE? Prove it or give a counterexample. i) If...

a-) Is the following statements TRUE or FALSE? Prove it or give a counterexample.

i) If f(x) : Rⁿ → R is a convex function, then for all α ∈ R, the set {x : f(x) ≤ α} is a convex set.

ii) If {x : f(x) ≤ α} is a convex set for all α ∈ R, then f(x) is a convex function.

b-) Prove that if x^* is a vector such that ∇g(x^* ) = 0 and ∇² g(x^*) is positive definite, then x^* is a local minimizer for g(x).

Expert Solution

Colby Messinger answered 5 months ago

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