Try Magic Notes and save time.Try it free
Try Magic Notes and save timeCrush your year with the magic of personalized studying.Try it free
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 $\mathbf{c}$ 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

Verified

Using 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.

## Recommended textbook solutions #### Calculus

9th EditionISBN: 9780131429246 (4 more)Dale Varberg, Edwin J. Purcell, Steve E. Rigdon
6,552 solutions #### Thomas' Calculus

14th EditionISBN: 9780134438986 (3 more)Christopher E Heil, Joel R. Hass, Maurice D. Weir
10,144 solutions #### Calculus: Early Transcendentals

8th EditionISBN: 9781285741550 (4 more)James Stewart
11,084 solutions #### Calculus: Early Transcendentals

9th EditionISBN: 9781337613927 (3 more)Daniel K. Clegg, James Stewart, Saleem Watson
11,049 solutions