Suppose that f(x,y) is a smooth function and that its partial derivatives have the values, f_x(-9, 9) = 2 and f_y(-9, 9) = 3. Given that f(-

There are an infinite number of possibilities!  However, perhaps the simplest is to assume f(x,y) takes the form a*x+b*y+c, where a, b and c are constants. f_x (the partial derivative of f with respect to x) is a, so a = 2; f_y is b, so b = 3, f(-9,9) is -4 so:

2*(-9)+3*(9)+c=-4  so c = -13.

Therefore f(-9,10) = 2*(-9)+3*(10)-13 = -1

and f(-8,9) = 2*(-8)+3*(9)-13 = -2

The first order approximation of f(x,y) around some point (a,b) is:

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

In your case:

$$f(x,y) = f(-9,9) + f_x(-9,9) (x - -9) + f_y(-9,9) (y - 9)$$

Fill in your values to get the results...