Prove the following version of the Mean-Value Theorem: If f
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Prove the following version of the Mean-Value Theorem: If f .x; y/ has first partial derivatives continuous near every p
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Prove the following version of the Mean-Value Theorem: If f .x; y/ has first partial derivatives continuous near every point of the straight line segment joining the points .a; b/ and .a C h; b C k/, then there exists a number satisfying 0< <1 such that f .a C h; b C k/ Df .a; b/ C hf1.a C h; b C k/ C kf2.a C h; b C k/: (Hint: Apply the single-variable Mean-Value Theorem to g.t / D f .a C th; b C tk/.) Why could we not have used this result in place of Theorem 3 to prove Theorem 4 and hence the version of the Chain Rule given in this section?
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