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Section 14.3 : Relative Minimums and Maximums

Find and classify all the critical points of the following functions.

1. \(f\left( {x,y} \right) = 2y - 9x - xy + 5{x^2} + {y^2}\)
2. \(f\left( {x,y} \right) = {x^3} - {y^3} + 8xy\)
3. \(f\left( {x,y} \right) = \left( {y - x} \right)\left( {1 - 2x - 3y} \right)\)
4. \(\displaystyle f\left( {x,y} \right) = \frac{1}{2}{x^4} - 4x{y^2} - 2{x^2} + 8{y^2}\)
5. \(f\left( {x,y} \right) = xy\,{{\bf{e}}^{ - 8\left( {{x^2} + {y^2}} \right)}}\)
6. \(\displaystyle f\left( {x,y} \right) = 8x - x\sqrt {y - 1} + {x^3} + \frac{1}{2}y - 12{x^2}\)