Unconstrained and constrained extrema

Find and classify the stationary points of $f(x,y)=x^2+y^2-4x+6y$.

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Find and classify the stationary points of $f(x,y)=2x^2-2xy+y^2-2x$.

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Find and classify the stationary points of $f(x,y)=x^3-12x+y^2$.

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Find and classify the stationary points of $f(x,y)=x^3+3xy-y^3$.

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Find and classify the stationary points of $f(x,y)=e^x(x+y^2)$.

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Verify that the origin is the only stationary point of $f(x,y)=(x-y{)}^2+y^4$, that at it the Hessian test does not allow a conclusion, and classify it.

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Find the absolute maximum and minimum of $f(x,y)=x^2+y^2-2x-2y$ on the square $[0,3]\times [0,3]$.

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Find the absolute maximum and minimum of $f(x,y)=x^2-y^2$ on the disk $x^2+y^2\le 4$.

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Find the maximum and minimum of $f(x,y)=x+y$ on the ellipse $x^2+4y^2=20$.

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Find the minimum of $f(x,y)=x^2+y^2$ on the line $2x+y=5$ and state whether $f$ has a maximum on the line.

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A monopolist sells two goods with inverse demands $p_1=40-q_1$ and $p_2=60-2q_2$ and has cost $C(q_1,q_2)=q_1^2+q_2^2+100$. Find the quantities, the prices and the maximum profit.

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A firm in perfect competition sells two goods at prices of $32$ and $22$ euros, with cost $C(q_1,q_2)=q_1^2+q_1{q}_2+q_2^2$. Find the quantities that maximize profit and the maximum profit.

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A consumer has utility $U(x,y)=x{y}^2$ and an income of $60$ euros, with prices $p_1=2$ and $p_2=4$ euros. Find the optimal bundle, with $x,y>0$, and the maximum utility.

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A firm has production function $Q=\sqrt{LK}$ and pays $9$ euros for labor and $4$ euros for capital per unit. Find the combination of factors that produces $Q=6$ at minimum cost and the minimum cost.

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