Let f : Rn → R be a differentiable function. Suppose that a point x∗ is...

Let f : Rn → R be a differentiable function. Suppose that a point x∗ is a local minimum of f along every line passes through x∗; that is, the function

g(α) = f(x^∗ + αd)

is minimized at α = 0 for all d ∈ R^n.

(i) Show that ∇f(x∗) = 0.

(ii) Show by example that x^∗ neen not be a local minimum of f. Hint: Consider the function of two variables

f(y, z) = (z − py^2)(z − qy^2),

where 0 < p < q.

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Expert Solution

Colby Messinger answered 1 year ago

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