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CSS Pure Mathematics QUESTION #1266
Question 1
For \(f(x,y)=xy-2x\) on the triangular region with vertices \((0,0)\), \((4,0)\), \((0,4)\), the absolute minimum value is:
  • \(-8\)✔️
  • \(0\)
  • \(-4\)
  • \(4\)
Correct Answer Explanation
On boundary edge from \((0,0)\) to \((4,0)\) (\(y=0\)): \(f=-2x\), minimum at \(x=4\): \(f=-8\). Interior critical point: \(f_x=y-2=0\Rightarrow y=2\); \(f_y=x=0\Rightarrow x=0\), giving \(f(0,2)=0\). Checking all boundaries, the absolute minimum is \(-8\) at \((4,0)\).