answer:To find the maximum value of x + y^2, we'll use the given constraint x^2 + y^2 = 4. We can rearrange this equation to express x^2 in terms of y^2: x^2 = 4 - y^2 Now, let's consider the expression x + y^2. We can rewrite x as √(4 - y^2) since x^2 = 4 - y^2. However, to avoid dealing with square roots, we can instead consider the square of the given expression: (x + y^2)^2 = x^2 + 2xy^2 + y^4 Substituting x^2 = 4 - y^2, we get: (x + y^2)^2 = (4 - y^2) + 2xy^2 + y^4 Expanding and rearranging, we have: (x + y^2)^2 = 4 + 2xy^2 - y^2 + y^4 = 4 + 2xy^2 + y^2(y^2 - 1) We want to find the maximum value of (x + y^2). Since the square root function is monotonically increasing, maximizing (x + y^2)^2 will maximize (x + y^2). Notice that y^2(y^2 - 1) is always non-positive for 0 ≤ y^2 ≤ 4, since 0 ≤ y^2 ≤ 1 makes y^2(y^2 - 1) ≤ 0 and 1 ≤ y^2 ≤ 4 makes y^2(y^2 - 1) ≥ 0 but in this case this term would lead to minimizing (x + y^2)^2. When y^2 = 1, y^2(y^2 - 1) = 0. So, we want y^2 = 1 to maximize (x + y^2)^2. When y^2 = 1, we have x^2 + y^2 = x^2 + 1 = 4, so x^2 = 3 and x = ±√3. Since x + y^2 is maximized when x is maximized, we choose x = √3. Thus, the maximum value of x + y^2 is √3 + 1.

answer:<tool_call>[{name: calculate_values, arguments: {num: 42.5}}]</tool_call>

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answer:Here is the Python implementation of the `normalize_coefficients` function: ```python def normalize_coefficients(a, b, c): def gcd(a, b): if a == 0: return b return gcd(b % a, a) g = gcd(a, b) x = a / g y = b / g z = c / g return (x, y, z) ``` The `normalize_coefficients` function takes three arguments `a`, `b`, and `c` representing the linear equation coefficients. It first calculates the GCD of `a` and `b` using the Euclidean algorithm. Then, it divides each coefficient by the GCD to get the normalized coefficients `x`, `y`, and `z`. The function returns these normalized coefficients as a tuple `(x, y, z)`. To test the `normalize_coefficients` function, you can use the following code: ```python x, y, z = normalize_coefficients(4, 8, 12) assert x == 1 and y == 2 and z == 3 ```