Find the value of the variable x1 that maximizes the functio

Find the value of the variable x_1 that maximizes the function f(x_1) =-(4 - x_1) + 2x_1 + 3 sure to check the second-order condition.

Solution

f(x₁) = -(4 - x₁)² + 2x₁ + 3

or

f(x₁) = -(4² + x_1² - 2*4*x₁) + 2x₁ + 3

       = -16 - x_1² + 8x₁+ 2x₁ + 3

       = - x_1² + 10x₁+ 3

For maximization, the first order condition is the first derivative should equal to Zero.

d(- x_1² + 10x₁+ 3) /dx₁ = 0

=> -2x₁ + 10 = 0

=> 2x₁ = 10

=> x₁ = 10/2 = 5

For maximization, the second order condition is the second derivative should less than Zero

d(-2x₁ + 10) / dx₁ = -2 (Note: -2 is less than zero

Therefore, at x₁ = 5 ,both first and second order condition is fulfilled.