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CSS Pure Mathematics QUESTION #1281
Question 1
For \(f(x,y)=2x^3+y^2-9x^2-4y+12x-2\), the critical points are found by setting:
  • Only \(f_x=0\)
  • Only \(f_y=0\)
  • \(f_x=0\) and \(f_y=0\) simultaneously✔️
  • \(f_{xx}=0\) and \(f_{yy}=0\)
Correct Answer Explanation
Critical points occur where all first partial derivatives vanish simultaneously: \(f_x=6x^2-18x+12=6(x^2-3x+2)=6(x-1)(x-2)=0\Rightarrow x=1\) or \(x=2\); \(f_y=2y-4=0\Rightarrow y=2\). Critical points: \((1,2)\) and \((2,2)\).