**Problem Statement** The temperature at a point \((x, y, z)\) on the unit sphere is defined by the function:\[ T(x, y, z) = 1 + xy + yz. \]**Task** Using the method of Lagrange multipliers, determine the temperature at the hottest point on the sphere.**Explanation** The problem requires finding the maximum value of the temperature function on the surface of a unit sphere. This is an optimization problem where the constraint is the equation of the unit sphere: \[ x^2 + y^2 + z^2 = 1. \]To solve this, the Lagrange multiplier technique is applied. This involves creating a Lagrangian function where the gradient of the temperature function is set proportional to the gradient of the constraint, leading to a system of equations to determine the critical points. From these, the maximum temperature is found.

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4. The temperature of a point (x, y, z) on the unit sphere is given by T(x, y, z) = 1 + xy + yz. By using the method of Lagrange multiplier find the temperature of the hottest point on the sphere.