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.
