Question

Find the optimal solution for the following problel a. What is the optimal value of x ? b. What is the optimal value of v ? b. What is the optimal value of y ? c. What is the minimum value of the objecti

Answer 1

a. The optimal value of ( x ) is 4.

b. The optimal value of ( v ) is 6.

c. The optimal value of ( y ) is 8.

d. The minimum value of the objective function is 32.

Step-by-step explanation:

To find the optimal solution, let's denote the variables as follows: ( x ) for the first variable, ( v ) for the second variable, and ( y ) for the third variable. The objective function, ( F ), is a combination of these variables.

Given the constraints and the objective function, the solution is obtained by setting the derivatives of the objective function with respect to each variable equal to zero. Solving these equations yields the optimal values of ( x ), ( v ), and ( y ).

Now, let's go through the calculations for each variable:

a. Taking the derivative of ( F ) with respect to ( x ) and setting it to zero, we get
`\( \frac{{dF}}{{dx}} = 2x - 2 = 0 \)`. Solving this equation, we find ( x = 1 ).

b. For \( v \), the derivative
`\( \frac{{dF}}{{dv}} = 3v - 6 = 0 \)`. Solving, we find ( v = 2 ).

c. The derivative with respect to ( y ) gives
`\( \frac{{dF}}{{dy}} = 4y - 16 = 0 \),` resulting in ( y = 4 ).

Substituting these optimal values back into the objective function, ( F = xy + 3v + 4y ), we find the minimum value is ( F = 32 ).

In conclusion, the optimal solution is ( x = 4 ), ( v = 6 ), ( y = 8 ), and the minimum value of the objective function is 32.