Example: Location of a Local Minimum, Maximum or Saddle Point

A flat plate is heated such that the temperature, T at any point (x, y) can be described as T = x^{2} + 2y^{2} – x. Find the temperature at the coldest point on the plate.

Solution

Take the partial derivatives and set them equal to zero:

x = 1/2

y = 0

Now plug these values back into the equation for temperature:

T(0.5, 0) = (0.5)^{2} + 2(0)^{2} – 0.5 = -0.25C

But is this a minimum (coldest temperature)? Calculate D to check. Find find the second partial derivatives.