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Exercises 34.3 Homework

1.

(Theoretical) Use the Lagrange multiplier method to find the critical points of the function \(f(x, y) = (x + 7y)^3 \) on the circle \(x^2 + y^2 = 1\text{.}\) Compare with the result of Example 34.1.2.

2.

Using the penalty method of Example 34.1.2, find four numbers \(x_1, x_2, x_3, x_4\) on the interval \([-1, 1]\) for which the product of all pairwise distances

\begin{equation*} \prod_{1\le k \lt \ell\le 4}|x_k - x_\ell| \end{equation*}

is as large as possible. What pattern do you observe in the points that achieve this maximum?

Remark: such maximization problem makes sense for any number of points, and on any bounded set. The points that maximize the product of distances are known as Fekete points and are used in approximation theory, specifically for polynomial interpolation.

Hint: to improve the performance of fminsearch, do not use absolute value: instead, minimize the product

\begin{equation*} \prod_{1\le k \lt \ell\le 4}(x_k - x_\ell) \end{equation*}

This product can be concisely expressed in Matlab as

prod(x(1:3)-x(4))*prod(x(1:2)-x(3))*(x(1)-x(2))

The constraint function c could be simply max(abs(x)) - 1; this constrains all four points to the interval at once.