Section 14.3 : Relative Minimums And Maximums - Assignment Problems Find and classify all the critical points of the following functions. 1. ( f(x, y)=2 y-9 x-x y+5 x^{2}+y^{2} ) 2. ( f(x, y)=x^{3}-y^{3}+8 x y ) 3. ( f(x, y)=(y-x)(1-2 x-3 y) ) 4. ( f(x, y)=frac{1}{2} x^{4}-4 x y^{2}-2 x^{2}+8 y^{2} ) 5. ( f(x, y)=x y mathbf{e}^{-8left(x^{2}+y^{2} ight)} ) 6. ( f(x, y)=8 x-x sqrt{y-1}+x^{3}+frac{1}{2} y-12 x^{2} )
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### For \( f(x, y) = (y - x)(1 - 2x - 3y) \): \[ f(x, y) = y - x - 2xy + 2x^2 - 3y^2 + 3xy \] \[ f_x = \frac{\partial f}{\partial x} = -1 - 2y + 4x + 3y \] \[ f_y = \frac{\partial f}{\partial y} = 1 - 3x - 6y \] Show more…
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In Exercises 5 - 14, find the critical points of the given function. Use the Second Derivative Test to determine if each critical point corresponds to a relative maximum, minimum, or saddle point. 5. f(x,y) = 1/2x^2 + 2y^2 - 8y + 4x 6. f(x,y) = x^2 + 4x + y^2 - 9y + 3xy 7. f(x,y) = x^2 + 3y^2 - 6y + 4xy 8. f(x,y) = 1 / (x^2 + y^2 + 1)
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