mutivariable unconstrained optimization using gradient search procedure

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Multi-variable unconstrained optimization problem:
Maximize the function,
$$f(x)=2xy+2y-x^2-2y^2$$ using the gradient search procedure.

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You can do the other method that your teacher wants, but this is slicker...$f(x,y) = -(x-y)^2 - (y-1)^2 + 1 \leq 1 \Rightarrow f_{\text{max}} = 1$ when $x = y = 1$.

Note: The trick is "completing square".