Differentiable function of two variables

Question

                                                                 

A differentiable function of two variables reaches its maximum value at (0,0) being the value of the function 5. A student calculates its Taylor polynomial, of second degree, at the point (0,0) and obtains the function

Can this polynomial be correct? Justify your answer.

Answer 1

P(x) doesn't look correct.

If a differentiable function f(x,y) has a maximum at (0,0), the first derivatives there must be 0.

However, the given Taylor polynomial shows that f_x(0,0) = 1 (the coefficient of x), which is not zero.