Prove multi-dimensional Mean Value Theorem

real-analysis calculus derivatives

Solution 1:

Everything is fine, except the last formula: You cannot divide by a vector. Leave it at $$f({\bf b})-f({\bf a})=\nabla f({\bf z})\cdot({\bf b}-{\bf a})$$ for some ${\bf z}\in[{\bf a},{\bf b}]$.

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