Surface Example -- Find Maximum by Hand

• Problem: Find the critical point of the surface defined by
   
  
  z = 13 + 2x + 3y - 2.5x² - 3.5y² + 2xy
  
  Verify that the point that you found is actually a maximum.
   
  
  1. Find the partial derivatives of z with respect to x and y, and set them to zero:
      
     
     ∂z/∂x = 2 - 5x + 2y = 0
     5x - 2y = 2          (Equation 1)
      
     ∂z/∂y = 3 - 7y + 2x = 0
     2x - 7y = -3       (Equation 2)
     
     
  2. Multiply Equation 2 by 2, multiply Equation 2 by -5, and add them to eliminate x:
      
     
      10x - 4y = 4
     -10x + 35y = 15
     ---------------------
      31y = 19
         y = 0.613
  3. Substitute the value of y obtained in 2 into Equation 1 to obtain x:
      
     
     5x - 2(0.613) = 2
     x = (2 + 2 * 0.613) / 5 = 0.645,
     
     Thus the critical point (x₀, y₀) is (0.645, 0.613).