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Question

Mean Value Theorem for Several Variables If f is differentiable at each point of the line segment from a to b, then there exists on that line segment a point c\mathbf{c} between a\mathbf{a} and b\mathbf{b} such that

f(b)f(a)=f(c)(ba)f(\mathbf{b})-f(\mathbf{a})=\nabla f(\mathbf{c}) \cdot(\mathbf{b}-\mathbf{a})

Assuming that this result is true, show that, if f is differentiable on a convex set S and if f(p)=0\nabla f(\mathbf{p})=\mathbf{0} on S, then f is constant on S. Note: A set S is convex if each pair of points in S can be connected by a line segment in S.

Solution

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Answered 2 years ago
Answered 2 years ago

Using the given theorem, let's consider two points, a\bold{a} and b\bold{b}. If we pick a point p\bold{p} on a line that connects points a\bold{a} and b\bold{b}, we can write:

f(a)f(b)=f(p)(ab)\begin{aligned} f(\bold{a})-f(\bold{b})&= \nabla f(\bold{p})(\bold{a}-\bold{b}) \end{aligned}

Considering the given condition:

f(a)f(b)=0(ab)f(\bold{a})-f(\bold{b})= \bold{0}\cdot(\bold{a}-\bold{b})

f(a)f(b)=0f(\bold{a})-f(\bold{b})= \bold{0}

meaning that the given function ff is constant on the given set.

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