Minimize the function subject to the constraint .

Question

Minimize the function  subject to the constraint  .

Quizplus · Accepted Answer

Using Lagrange multiplier, the Lagrangian function is:
L = 11eaa8b1_d995_c43e_b1b6_512deb2bb09a_TB6026_11 - λ(11eaa8b1_d995_c43f_b1b6_fb68e75c13f9_TB6026_11)
The partial derivative with respect to aa8b1_d995_c43e_b1b6_512deb2bb09a_TB6026_11 gives:
11 - λ = 0 
This gives λ = 11 
Substituting this back in the Lagrangian function, the constraint equation simplifies to:
11eaa8b1_d995_c43e_b1b6_512deb2bb09a_TB6026_118 = 11eaa8b1_d995_c43f_b1b6_fb68e75c13f9_TB6026_118 
Simplifying gives:
eaa8b1_d995_c43e_b1b6_512deb2bb09a_TB6026_118 = eaa8b1_d995_c43f_b1b6_fb68e75c13f9_TB6026_118 
Therefore, the given function reduces to a function of one variable, eaa8b1_d995_c43e_b1b6_512deb2bb09a_TB6026_118:
11eaa8b1_d995_c43e_b1b6_512deb2bb09a_TB6026_118 - 11λ = f(eaa8b1_d995_c43f_b1b6_fb68e75c13f9_TB6026_118) 
11eaa8b1_d995_c43e_b1b6_512deb2bb09a_TB6026_118 - 11(11) = f(eaa8b1_d995_c43f_b1b6_fb68e75c13f9_TB6026_118) 
eaa8b1_d995_c43e_b1b6_512deb2bb09a_TB6026_118 - 121 = f(eaa8b1_d995_c43f_b1b6_fb68e75c13f9_TB6026_118) 
The minimum value of f(x) is 0, which occurs when x = 1. Therefore, the minimum value of the given function is:
eaa8b1_d995_c43e_b1b6_512deb2bb09a_TB6026_118 - 121 = 1 - 121 = -120 
Thus, the answer is (D).

Minimize the Function Subject to the Constraint