<s>[INST] Can anyone tell me if this is correct? Suppose that the temperature of a metal plate is given by $T(x; y) = x^2 +2x+y^2$, for points $(x, y)$ on the elliptical plate defined by $x^2 + 4y^2 <= 24$.
Find the maximum and minimum temperatures on the plate.
This is what i have done so far.
Finding critical point:
$T(x)=2x+2$, $T(y)=2y$. Equating to $0$, $x=-1$, $y=0$.
Critical point is $(-1,0)$ and is a minimum.
On the boundary,   $  x^2 + 4y^2 = 24$
$g(x,y)=x^2 + 4y^2$
$g(x)=2x, g(y)=8y$
$2x+2=A2x$---------(1)
$2y  =A8y$---------(2)
$x^2+4y^2=24$------(3)
When solving from equation 1 and 3 im getting $ x=-1,y=|(23/4)^{0.5}|,$ and $x=|24^{0.5}|, y=0$ and when from eqn 2 and 3 im getting $x=|24^{0.5}|,y=0, A= 0.25 , x=-4/3, y=|(50/9)^{0.5}|$. 
Is this correct? am getting different values when using equation $1$ and $2$.
 [/INST] For minimum, point $(-1,0)$ should be correct.
Although I didn't understand your approach for finding maximum, maximum temperature will occur at the boundaries.
$x^2 + 4y^2 = 24$
$y^2 = \frac{24-x^2}{4}$
Substituting this value into the equation for temperature yields $T(x)= \frac{3x^2+8x+24}{4}$
As $x$ ranges from $-\sqrt{24} \space to \space \sqrt{24}$.
$x=\sqrt{24} $ gives $33.798$ as the maximum.
</s>