Algebra

First, let's combine some like terms to simplify this expression!

$x^2 -14x + y^2 + 16y + 26$

Now, all we have to do is to take the derivative with respect to x and y and set that to 0 before plugging the values back into the function. 

We have 

$2x - 14 = 0 \\ 2x = 14 \\ x = 7$

and 

$2y + 16 = 0 \\ 2y = -16 \\ y = -8$

Now, plugging the value back into the function, we have $(7)^2 - 14(7) + (-8)^2 + 16(-8) + 26 = -87$. 

So the lowest temperature is just -87!

Thanks! :)