Question

Maximum value of the function f(x, y) = 2 + 2x + 2y – x2 - y2 is​

Answer 1

Answer:

this os a correct answer

Answer 2

f(x , y) = 2 + 2x + 2y - x² - y² is maximum at (1 , 1)  and maximum value is 4

Given:

• f(x , y) = 2 + 2x + 2y - x² - y²

To Find:

• Maximum value of the function

Solution:

f(x , y) = 2 + 2x + 2y - x² - y²

Step 1:

Rewrite the function by rearranging the terms:

f(x , y) = 2 + 2x + 2y - x² - y²

=> f(x , y) =2  -(x² - 2x)  - (y² - 2y)

=>  f(x , y) =2  -(x² - 2x+ 1) + 1 - (y² - 2y + 1) + 1

=>  f(x , y) =4  -(x² - 2x+ 1)   - (y² - 2y + 1)

Step 2:

Use the identity (a - b)² = a² - 2ab + b²

f(x , y) =  4 - (x - 1)² - (y - 1)²

Step 3:

Square of a term is always non negative  Hence f(x, y) is maximum when

(x - 1)² and  (y - 1)² = 0

f(x , y) =  4 - 0 - 0  = 4

x = 1 , y = 1

f(x , y) = 2 + 2x + 2y - x² - y² is maximum at (1 , 1)  and maximum value is 4