please proof all parts step by step to get all credit Let f : Rn rightarrow R be
ID: 2981651 • Letter: P
Question
please proof all parts step by step to get all credit
Let f : Rn rightarrow R be differentiable, and gamma a differentiable path such that f(gamma(t)) is constant. Prove the f(gamma(t)) is orthogonal to gamma'(t) for all t. Suppose that f : Rn rightarrow R is differentiable at a alpha Rn. The rate of growth of f in the direction upsilon Rn is given by the directional derivative D upsilon f(a). Show that direction of maximal growth, i.e. the unit vector upsilon for D upsilon f(a) is maximal, is f(a)/| f(a)|. Interpret (i) and (ii) in terms of the following scenario: you are hiking in mountainous terrain, and f(x,y) represents the height of the terrain. If you are in possession of a map that includes contours lines, how should you walk if you want to reach a nearby summit as quickly as possible? Illustrate your argument using a sketch.Explanation / Answer
If Hf is negative definite at x, then f attains a strict local maximum at x iff
5f (x) = 0
2 In (1), replace
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