Global Max and Min Hi can someone please help me figure out how to start this question?
Temperature within a circular region $R = \{(x, y)| x^2 + y^2 \le 49\}$ is given by $T(x,y) = 4x^2 -4xy + y^2$
Find the global minimum and maximum temperatures in the region and the points where these extreme temperatures occur.
I'm supposed to use Lagrange multipliers and parametrization which I don't really understand. If someone can please help me get started I would really appreciate it.

We use Lagrange multipliers. Let, $F(x,y) = 4x^2 - 4xy + y^2 + \lambda(x^2 + y^2 - 49)$. Thus, 
$$
\begin{cases}
\dfrac{\partial F}{\partial x} = 8x - 4y + 2\lambda x = 0 \qquad (1)\\
\dfrac{\partial F}{\partial y} = 8y - 4x + 2\lambda y = 0 \qquad(2)\\
\dfrac{\partial F}{\partial \lambda} = x^2 + y^2 -49= 0 \qquad (3)
\end{cases}
$$ 
From equations (1) and (2), we have,
$$
\dfrac{2x  - 4y}{y} = \lambda = \dfrac{2y -4x}{x} \quad \Rightarrow \quad  x^2 = y^2
$$
Thus, from equation (3), $2x^2 = 49 \ \Rightarrow \ x = \pm \frac{7}{\sqrt{2}}$ and $y = \pm \frac{7}{\sqrt{2}}$
The points of maximum and minimum are $(-\frac{7}{\sqrt{2}},-\frac{7}{\sqrt{2}})$ and $(\frac{7}{\sqrt{2}},\frac{7}{\sqrt{2}})$.
