question is: Identify the functions z=2x^2-y^2 extrema under the condition that x+y=2

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You need to find the extreme values of `z=f(x,y)= 2x^2-y^2`  subject to the constraint`g(x,y)=x+y=2` , hence, you should use the method of Lagrange's multipliers, thus, you should find the values of `x,y,z,lambda`  that satisfy the equations x+y-2=0 and `gradf=lambda*grad g.`

`grad f = f_x*i + f_y*j`

`grad f = 4x*i - 2y*j`

`grad g = g_x*i + g_y*j`

`grad g = i + j`

You need to solve the equation `gradf=lambda*grad g`  such that:

`4x*i - 2y*j = lambda*i + lambda*j`

You need to compare the components such that:

`4x = lambda`

`-2y = lambda`

`4x = -2y =gt 2x = -y`

You should substitute `2x = -y`  in equation g(x,y)=0

`x+y = 2`

`x - 2x = 2 =gt -x = 2 =gt x = -2`

`y = 4`

Hence, the function has an extreme at (-2,4) and the extreme value is f(-2,4)= -8.