Question
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Answer
The minimum value of
is
when
and
.
Solution
To minimize the function
, we need to consider the constraints that define the feasible region for
and
. However, since no constraints are provided in your problem, I will assume that
and
can take any non-negative values (i.e.,
and
).
-
Identify the function to minimize:
-
Determine the behavior of the function:
- The function
is linear, and its minimum value occurs at the boundary of the feasible region. - Since both coefficients (6 and 7) are positive,
decreases as and decrease.
- The function
-
Evaluate the minimum:
- The minimum value occurs at the point where both
and are at their lowest values, which is and .
- The minimum value occurs at the point where both
-
Calculate the minimum value:
Thus, the minimum value of
is
at the point
.
If there are specific constraints you would like to include, please provide them for a more detailed analysis.
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Bonus Knowledge
To minimize the objective function
, you’re generally solving an optimization problem, often seen in linear programming. The key here is to define constraints that limit the values of
and
. Once you have your constraints, you’d typically graph the feasible region, identify corner points, and evaluate
at those points to find the minimum.
For a real-world application, say you’re managing resources in a factory, where
might represent the number of product A made, and
the number of product B. By minimizing
, you’re reducing costs while ensuring you meet production requirements. This approach can help enhance profit margins by avoiding unnecessary expenditure.