Final answer:
To minimize the expression x^2 - 2y^2 given that x + y = 24, you express y in terms of x, substitute it into the original expression, and then use differentiation to find the critical point that will give the minimum value, ensuring x and y are non-negative.
Step-by-step explanation:
To find the values of x and y that minimize the expression x2 - 2y2 given that x + y = 24 and both are non-negative numbers, we can use calculus or reason through the symmetry of the equation. Since we're dealing with a minimization problem of a quadratic expression, and we have a linear constraint, this is a perfect candidate for the method of Lagrange multipliers or completing the square. However, for a high school level approach, we can make do with a simpler method.
First, express y in terms of x using the equation y = 24 - x. Then, substitute for y in the original expression to get a quadratic equation in terms of x only. Now, we simplify the equation: x2 - 2(24 - x)2.
Upon expanding the squared term and simplifying, we can differentiate the resulting expression with respect to x to find the critical points and then test these points to find the minimum value. Alternatively, since the second derivative will be positive (a cup-upward parabola), the critical point will give us the minimum value of the quadratic function.
Remember to check that the values of x and y are non-negative, as per the constraints of the problem. The final step would be to find the corresponding value of y once the minimizing value of x is determ