A flat metal plate is mounted on a coordinate plane. The temperature of the plate, in degrees Fahrenheit, at point $(x, y)$ is given by $x^{2}+3 y^{2}-4 x+6 y$. Find the minimum temperature and where it occurs. Is there a maximum temperature? Determine the minimum temperature and its location. Select the correct choice below and fill in any answer boxes within your choice. A. The minimum temperature is $\square^{\circ} \mathrm{F}$ at $(x, y)=\square$. (Simplify your answers.) B. There is no minimum temperature. Determine the maximum temperature and its location. Select the correct choice below and fill in any answer boxes within your choice. A. The maximum temperature is $\int^{\circ} \mathrm{F}$ at $(x, y)=$ (Simplify your answers.) B. There is no maximum temperature.

Step 1 ：The given function is a quadratic function in two variables. The minimum or maximum of a quadratic function occurs at its vertex. The vertex of a quadratic function given in the form \(f(x, y) = ax^2 + by^2 + cx + dy + e\) is at the point \((-\frac{c}{2a}, -\frac{d}{2b})\). In this case, \(a = 1\), \(b = 3\), \(c = -4\), and \(d = 6\). So, the vertex is at \((\frac{4}{2}, -\frac{6}{2*3}) = (2, -1)\).

Step 2 ：The temperature at this point is \(f(2, -1) = 2^2 + 3*(-1)^2 - 4*2 + 6*(-1)\).

Step 3 ：The minimum temperature is -7 degrees Fahrenheit and it occurs at the point (2, -1).

Step 4 ：Since the coefficients of \(x^2\) and \(y^2\) are positive, the function opens upwards and hence there is no maximum temperature.

Step 5 ：The minimum temperature is \(\boxed{-7^{\circ} \mathrm{F}}\) at \((x, y)=\boxed{(2, -1)}\). There is no maximum temperature.