The corner points of the feasible region determined by a system of linear constraints are $(0,3),(1,1)$ and $(3,0)$. If the objective function is $z = px+qy$, where $p,q > 0$, then the condition on p and q such that the minimum of z occurs at both $(3,0)$ and $(1,1)$ is _____

Question

The integral $ I = \int_{\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x + \frac{\pi}{4}}{2 - \cos 2x} \, dx $ is equal to:

Answer 1

Step 1: Use the slope of the iso-profit line:
If the minimum is at two vertices, the line $px+qy=z$ must be parallel to the edge joining $(3,0)$ and $(1,1)$.

Step 2: Edge slope:
Slope $=\dfrac{1-0}{1-3}=-\dfrac12$. The slope of $px+qy=z$ is $-\dfrac pq$. So $\dfrac pq=\dfrac12$, giving $p=\dfrac q2$.

Step 3: Pick:
Option C.

Final Answer:
Equal z values at (3, 0) and (1, 1) give 3p = p + q. \[ \boxed{\text{(C) }p=\dfrac q2} \]

The corner points of the feasible region determined by a system of linear constraints are \((0,3),(1,1)\) and \((3,0)\). If the objective function is \(z = px+qy\), where \(p,q > 0\), then the condition on p and q such that the minimum of z occurs at both \((3,0)\) and \((1,1)\) is _____

Show Hint

The Correct Option is C

Solution and Explanation

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