Consider the functions

f(x1; x2) = x1x2

g(x1; x2) = ax1x2

h(x1; x2) = ln(ax1x2); where a > 0 is an unknown constant and we are only interested in points where x1 > 0 and x2 > 0.

(a) Calculate all partial derivatives of f(x1; x2), g(x1; x2), and h(x1; x2).

(b) Calculate the slopes of the level curves of f(x1; x2), g(x1; x2), and h(x1; x2) at an arbitrary point (x1; x2).

(c) Extra credit: Comment on your calculation in (b). How can we interpret the results? What happens with the level curves of a function if we multiply the function with a positive constant or take the natural logarithm of it? Why?

(d) Consider the function v(x1; x2) = 2x1 + x2. Calculate the point (x 1 ; x 2 ) at which the level curve of f(x1; x2) is tangent to the level curve defined by v(x1; x2) = 10. Hint: Proceed in three steps:

1. Calculate the slope of the level curve of v(x1; x2).

2. Write down a system of two equations. The first equation sets the slopes of the level curves of f(x1; x2) and v(x1; x2) equal to each other. The second equation is v(x1; x2) = 10.

3. Solve the system of equations for x1 and x2.