Solution of IDZ Ryabushko 10.2 - Option 30

Section: Chapter 10. Differential calculus of functions of several variables
Topic: Applications of differential calculus of functions of several variables
Description: Finding equations of the tangent plane and normal to a surface, computing second-order partial derivatives, checking the fulfillment of differential equations (e.g., Laplace's equation), investigating the extremum of a function of two variables, finding the largest and smallest values of a function in a closed domain.

Tasks:
1. Find the equations of the tangent plane and the normal to the given surface S at the point M₀(x₀, y₀, z₀)
2. Find the second-order partial derivatives of the indicated functions. Verify that z″xy = z″yx
3. Check whether the given function u satisfies the indicated equation.
4. Investigate the following functions for extremum.
5. Find the largest and smallest values of the function z = z(x, y) in the region D bounded by the given lines.

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