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\$.

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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!

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