Example

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.

   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.




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