## Related questions with answers

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 $\mathbf{c}$ between $\mathbf{a}$ and $\mathbf{b}$ such that

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

Assuming that this result is true, show that, if f is differentiable on a convex set S and if $\nabla f(\mathbf{p})=\mathbf{0}$ 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

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

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

Considering the given condition:

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

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

meaning that the given function $f$ is constant on the given set.

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