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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Critical Thinking

  • question_answer
    If \[u(x,y)=y\log x+x\,\log y,\] then \[{{u}_{x}}{{u}_{y}}-{{u}_{x}}\log x-{{u}_{y}}\log y+\log x\,\,\log y=\] [EAMCET 2003]

    A) 0

    B) -1

    C) 1

    D) 2

    Correct Answer: C

    Solution :

    • \[u(x,y)=y\log x+x\log y\]                   
    • \[{{u}_{x}}=\frac{y}{x}+\log y;\ {{u}_{y}}=\log x+\frac{x}{y}\]                   
    • Now, \[{{u}_{x}}{{u}_{y}}-{{u}_{x}}\log x-{{u}_{y}}\log y+\log x\log y\]               
    • \[=\left( \frac{y}{x}+\log y \right)\text{ }\left( \log x+\frac{x}{y} \right)-\frac{y}{x}\log x-\log x\log y-\log x\log y\] \[-\frac{x}{y}\log y+\log x\log y=1\].


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