Find the critical point of \(f(x,y)=x^2+y^2\).

Find all critical points of \(f(x,y)=x^2+y^2-4x+6y\).

Classify the critical point \((0,0)\) of \(f(x,y)=x^2+y^2\).

Classify the critical point \((0,0)\) of \(f(x,y)=x^2-y^2\).

Find and classify the critical point of \(f(x,y)=x^2+4y^2-2x+8y\).

Find the critical points of \(f(x,y)=xy-x-y\).

Classify the critical point of \(f(x,y)=xy-x-y\).

Find the critical points of \(f(x,y)=x^3-3x+y^2\).

Classify the critical points of \(f(x,y)=x^3-3x+y^2\).

For \(f(x,y)=x^4+y^4\), explain why \((0,0)\) is a local minimum even though the second derivative test is inconclusive.