Question

a) If you add the terms ax^2+bxy+cy^2 to the function L(x,y) = 1 - 0.5y, how would this affect the derivatives of L(x,y) at (0,0)?

b) What a, b and c you would pick to make those derivatives match the derivatives of the f(x,y) = sqrt(x^2+1-xy-y) at (0,0).

c) Define this new L(x,y).

Answer 1

The terms to be added to the given function are all either squares of one variable or a product of two variables, i.e., the entire term being added is a polynomial of second degree in two variables x,y, with only second degree terms of both those variables. So, any derivative of this term will be a polynomial of degree 1 in x,y, including only degree 1 terms, i.e., constant multiples of x,y. Obviously, putting x=y=0 will result in the entire term becoming 0, and therefore not affect that derivative of the original term in any way.

This was considering only first order derivatives of the functions.

b) In case the equality of derivatives referred to higher ones too, we compute the second order derivative of f, L and set them to be equal at (0,0).

Similarly, we set and to find b,c.

From the obtained information:

c)