Answer: To find the minimum value of c = x + 3y, you need to solve the optimization problem with a constraint or without any constraint.
Without a constraint:
The minimum value of c = x + 3y without a constraint is achieved when the partial derivatives of c with respect to x and y are equal to zero.
Let's take the partial derivative of c with respect to x:
dc/dx = 1
And with respect to y:
dc/dy = 3
Setting these equal to zero, we have:
1 = 0 and 3 = 0
Since the partial derivatives are not equal to zero, there is no minimum value of c without a constraint.
With a constraint:
If there is a constraint, such as a limit on x or y, then the minimum value of c can be found by using optimization techniques, such as the method of Lagrange multipliers.
Explanation: