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?


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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 $f"(x_0) = 0$


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