So far I have found the critical point $(0, 0)$ and the partials of $-5x^4$ and $4y^3$. How would I proceed after finding the critical point, since I don't have a definite interval. I get confused without an interval.
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$\begingroup$ You're talking about $z(x,y)=y^4-x^5$, right? $\endgroup$– Lucas PelizzariCommented Oct 9, 2022 at 1:29
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$\begingroup$ yeah, its a multivariable function. $\endgroup$– maria guallpaCommented Oct 9, 2022 at 1:32
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1 Answer
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The critical points of $f(x,y)=y^4-x^5$ are:
- All the points in the border
- All the points with $\nabla f=(0,0)$
Since there is no border, the critical points are just the last indicated, so: $$ \nabla f=(-5x^4,4y^3)=(0,0) \implies (x,y)=(0,0) $$
Hence the function has only one critical point, which turns to be only a saddle point (why?).
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1$\begingroup$ I turned your $\Delta f$ (Laplacian) into $\nabla f$ (gradient), as that is pretty clearly what you intended. Notation:
\nablafor $\nabla$. $\endgroup$ Commented Oct 9, 2022 at 2:52