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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