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Question: where d is the derivative...

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Δ Assume U C Fn is an open subset and f e C2(U; R)-Prove that if XoEU is such that Df (xo)-0 and D2f (xo) is positive definite (see Definition 4.5.1), then f attains a local minimum at xo. You may assume that there is a neighborhood B(xo,6) of Xo where every eigenvalue is greater than ε, and hence vD2f(x)v > 태vll for all v and for all x in B(xo, δ) (this follows from the implicit function theorem, which is proved in the next chapter). Hint Consider using Taylors theorem.

where D is the derivative

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