Final answer:
By expressing x and z in terms of y from the given constraints and substituting into the function f(x,y,z), we can solve for the maximum value. The maximum value of the function is -4, which occurs when x = 2, y = 4, and z = -4.
Step-by-step explanation:
To maximize the function f(x,y,z)=x2 + 2y - z2 subject to the constraints 2x - y = 0 and y + z = 0, we can solve the constraints to express x and z in terms of y. From the first constraint, 2x - y = 0, we can find that x = y/2.
From the second constraint, y + z = 0, we can find that z = -y. Now we can substitute these into the function to get f(y) = (y/2)2 + 2y - (-y)2, which simplifies to f(y) = y2/4 + 2y - y2.
To find the maximum value, we take the derivative of f(y) with respect to y and set it to zero. The derivative f'(y) = y/2 + 2 - 2y, when solved, gives y = 4.
Substituting y = 4 back into x = y/2 and z = -y, we get x = 2 and z = -4.
Finally, substituting these values into f(x,y,z), we get the maximum value of the function, which is f(2,4,-4) = 22 + 2(4) - (-4)2 = 4 + 8 - 16 = -4.