Say we have some arbitrary function $f(x,y) = xy$ or whatever. It doesn't have to be a scalar-valued function (though for my question it might be a restriction, not sure. Which is why I'm asking). If a problem asks to find the maximum and minimum values attained by that function along a certain path, what do I need to do? I know there are maximums and minimums when the derivative of a function is equal to $0$, but what does it mean when you find maximums and minimums along a path? Mathematically, how do you express a function along a path?
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Write the function in terms of the path then find the extrema. – Joseph Zambrano Mar 19 '14 at 16:29
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My question is how to write a function in terms of the path. For a more concrete example, $f(x,y)=xy$ along $\textbf{c}(t) = (cos t, sin t)$. – mr eyeglasses Mar 19 '14 at 16:33
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A path in $\mathbb R^2$ is a continuous function $\phi : [a,b] \to \mathbb R^2$. Finding maxima and minima of a function $f$ along a path $\phi$ means finding maxima and minima of the function $f \circ \phi$.
For example if $f(x, y) = xy$ and $\phi(t) = (\cos(t), \sin(t))$, with $t \in [0, 2 \pi]$, then to find the maxima and minima of $f$ along $\phi$ (the unit circle) you would find the maxima and minima of $$(f \circ \phi)(t) = \cos(t) \sin(t), \qquad t \in [0, 2\pi]$$
I know there are maximums and minimums when the derivative of a function is equal to 0
This is not true, consider $f(x) = x^3$ at $x = 0$. What is true is that if $f$ is differentiable at $x_0$ and $x_0$ is a local extremum then $f'(x_0) = 0$
dani_s
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