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  3. let f : rn ? r be a differentiable function. suppo

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

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Advance Math

Let f : Rn ? R be a differentiable function. Suppose that apoint x? is a local minimum of f along every line passes throughx?; that is, the function

g(?) = f(x^? + ?d)

is minimized at ? = 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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