Free and constrained maxima and minima

Find and classify the critical points of the function

$$f(x,y)=x^2+xy+y^2-3x.$$

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Find and classify the critical points of the function

$$f(x,y)=x^4+y^4-4xy.$$

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Find and classify the critical points of the function

$$f(x,y)=(x-y{)}^2+y^3-3y.$$

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Find all the critical points of the function

$$f(x,y)=x^2{y}^2$$

and determine which of them are extremum points.

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Find and classify the critical points of the function

$$f(x,y)=x^2+y^4-2x.$$

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Find the absolute maximum and minimum of the function

$$f(x,y)=x^2-y^2\text{on }K=[-1,2]\times [0,1].$$

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Find the absolute maximum and minimum of the function

$$f(x,y)=x^2+y^2-2y\text{on the disk }{x}^2+y^2\le 4.$$

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Find the absolute maximum and minimum of the function

$$f(x,y)=x+y-xy$$

on the triangle with vertices $(0,0)$, $(2,0)$, $(0,2)$, boundary included.

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Find the maximum and minimum of the function

$$f(x,y)=2x+y\text{subject to the constraint }{x}^2+y^2=20.$$

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Find the points of the curve $xy=4$ at minimum distance from the origin and compute that distance.

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Find the maximum and minimum of the function

$$f(x,y,z)=x+y+z\text{on the sphere }{x}^2+y^2+z^2=12.$$

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Among the rectangles with area $36$, find the one with minimum perimeter.

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A box with a square base and no lid must have volume $4{\mathrm{dm}}^3$. Find the dimensions that minimize the surface area (base and four walls).

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