Find the linearization L(x,y) of the function f(x, y) = (x + y + 2)2 at the point (1, 2)

The linearization of a function at a given point is given by the formula:

L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)

where (a, b) is the point at which we are linearizing, f_x and f_y are the partial derivatives of f with respect to x and y, respectively.

Given the function f(x, y) = (x + y + 2)^2, we first need to find its partial derivatives.

The partial derivative of f with respect to x is:

f_x(x, y) = 2(x + y + 2)

And the partial derivative of f with respect to y is:

f_y(x, y) = 2(x + y + 2)

Now we can substitute the point (1, 2) into these expressions to find the values of the partial derivatives at this point:

f_x(1, 2) = 2(1 + 2 + 2) = 10 f_y(1, 2) = 2(1 + 2 + 2) = 10

Next, we substitute the point (1, 2) into the original function to find its value at this point:

f(1, 2) = (1 + 2 + 2)^2 = 25

Finally, we can substitute all these values into the formula for the linearization to find:

L(x, y) = 25 + 10(x - 1) + 10(y - 2)

So, the linearization of the function at the point (1, 2) is L(x, y) = 25 + 10x - 10 + 10y - 20 = 10x + 10y - 5.