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 = x2 + 2y2 – x. Find the temperature at the coldest point on the plate.


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.


Then since D>0 and >0, it is a minimum point.

Practice Problems