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Answer by coffeemath for Proof or counter-example that $(0,t_0)$ is a maximum...
This example is built around a standard example of a function for which both partials exist at a point, yet the function behaves badly near the point. That will be called $q(x,y),$ and is...
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Answer by paf for Proof or counter-example that $(0,t_0)$ is a maximum of $f:...
Let's build $f$ such that $g(t) = t-t^2$ (which attains its maximum at $t_0 = \dfrac12$) but which increases with $x$.For this purpose, let's try $f(x,y) = x + y - y^2$. When we fix a value for $x$,...
View ArticleProof or counter-example that $(0,t_0)$ is a maximum of $f: [0,1]^2 \to...
Suppose $f: [0,1] \times [0,1] \to \mathbb{R}$ is a differentiable function. Define $g(t) = f(0,t)$ and suppose that $g$ has a maximum in $t_0 \in (0,1)$, and suppose additionally that $D_1f(0,t_0)...
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