Say we have actually some arbitrary feature $f(x,y) = xy$ or whatever. It does not need to be a scalar-valued attribute (though for my question it can be a restriction, not sure. Which is why I"m asking). If a difficulty asks to discover the maximum and minimum worths attained by that attribute alengthy a certain route, what perform I should do? I recognize tbelow are maximums and also minimums when the derivative of a function is equal to $0$, yet what does it expect when you discover maximums and also minimums alengthy a path? juniorg8.comematically, just how carry out you expush a role along a path?


A path in $juniorg8.combb R^2$ is a continuous attribute $phi : o juniorg8.combb R^2$. Finding maxima and minima of a duty $f$ along a route $phi$ indicates finding maxima and also minima of the attribute $f circ phi$.

You are watching: Find the maximum and minimum values attained by the function f along the path c(t).

For instance if $f(x, y) = xy$ and also $phi(t) = (cos(t), sin(t))$, with $t in <0, 2 pi>$, then to discover the maxima and minima of $f$ along $phi$ (the unit circle) you would discover the maxima and minima of $$(f circ phi)(t) = cos(t) sin(t), qquad t in <0, 2pi>$$

I understand tbelow are maximums and also minimums as soon as the derivative of a function is equal to 0

This is not true, take into consideration $f(x) = x^3$ at $x = 0$. What is true is that if $f$ is differentiable at $x_0$ and $x_0$ is a regional extremum then $f"(x_0) = 0$


Thanks for contributing a response to juniorg8.comematics Stack Exchange!

Please be sure to answer the question. Provide details and share your research!

But avoid

Asking for assist, clarification, or responding to other answers.Making statements based on opinion; ago them up through recommendations or personal endure.

Use juniorg8.comJax to format equations. juniorg8.comJax reference.

See more: Ear Rhymes With Ear ' - Words And Phrases That Rhyme With Ear

To learn even more, check out our tips on creating good answers.

Post Your Answer Discard

By clicking “Article Your Answer”, you agree to our terms of business, privacy plan and also cookie policy

Not the answer you're looking for? Browse various other concerns tagged multivariable-calculus or ask your own question.

Does a summation of two negative Gaussians via various traditional deviations have actually a stationary point?
Find the absolute min and also max values taken by the feature $f(x,y)=2x^2+3y^2-4x-5$ on the disk $x^2+y^2leq 16$

website design / logo © 2021 Stack Exadjust Inc; user contributions licensed under cc by-sa. rev2021.9.10.40187

juniorg8.comematics Stack Exadjust functions best via JavaScript permitted

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange deserve to keep cookies on your gadget and discshed information in accordance through our Cookie Policy.