If x and y satisfy x^2+y^2=7, find the maximum value of x^2+2y^2+2x-4

To find the maximum value of the expression x^2 + 2y^2 + 2x - 4, we need to use the given condition x^2 + y^2 = 7. Let's go through the steps to solve this problem:

1. Start by expressing the expression we want to maximize in terms of a single variable, either x or y. In this case, let's solve for y in terms of x using the first equation:

x^2 + y^2 = 7
2y^2 = 7 - x^2
y^2 = (7 - x^2)/2
y = ±√((7 - x^2)/2)

2. Substitute the obtained value of y into the expression we want to maximize:

f(x) = x^2 + 2[(7 - x^2)/2] + 2x - 4
= x^2 + 7 - x^2 + 2x - 4
= 7 + 2x - 4
= 2x + 3

3. Differentiate the function f(x) with respect to x to find the critical points. Set the derivative equal to zero and solve for x:

f'(x) = 2
2 = 0
x = -3/2

4. Calculate the second derivative of the function f(x) to determine the nature of the critical point found:

f''(x) = 0 (since the derivative of a constant is zero)

5. Since the second derivative is zero, we need to examine the behavior of f(x) around the critical point -3/2. We can check the signs of f''(x) on either side of this point:

When x < -3/2, f''(x) < 0 (concave down)
When x > -3/2, f''(x) > 0 (concave up)

6. From the signs of the second derivative, we can conclude that x = -3/2 corresponds to a local minimum of f(x).

7. Since the domain of x is not specified, we also need to check the endpoints of the valid range of x, which depends on the condition x^2 + y^2 = 7.

When x = ±√7, y = 0, and f(x) = 2x + 3. Therefore, the maximum value occurs at x = √7 or x = -√7.

To summarize, the maximum value of x^2 + 2y^2 + 2x - 4 is obtained when x = ±√7, and the maximum value is 2(√7) + 3 or 2(-√7) + 3.