Maximum of a Linear Function subject to Linear and Quadratic Constraints in \( \mathbb{R}^4 \)

Calculus Level 4

Let \( \mathbf{x} = [ x_1, x_2, x_3, x_4 ]^T \) be a vector in \( \mathbb{R}^4 \), and let \( f( \mathbf{x} ) = 2 x_1 - 3 x_2 + 5 x_3 + 4 x_4 \) be the objective function to be maximized over \( \mathbb{R}^4 \), subject to the linear constraint, \( x_1 + 2 x_2 - x_3 + 3 x_4 = 10 \), and the quadratic constraint, \( x_1^2 + 2 x_2^2 + 3 x_3^2 + 4 x_4^2 + 2 x_1 + x_2 + 4 x_3 + x_4 = 100 \).

If \( (a, b, c, d) \) is the point in \( \mathbb{R}^4 \) at which the function \( f \) attains its maximum and, if the value of that maximum is \( f^* \), then find \( a + b + c + d + f^* \). Round your answer to 3 decimal places.

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