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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Maxima and Minima

  • question_answer
    The minimum value of \[[(5+x)(2+x)]/[1+x]\] for non-negative real x is                  [Kurukshetra  CEE 1998]

    A)            12

    B)            1

    C)            9

    D)            8

    Correct Answer: C

    Solution :

               Given \[f(x)=\frac{[(5+x)(2+x)]}{[1+x]}\]                    \[f(x)=1+\frac{4}{1+x}+(5+x)=(6+x)+\frac{4}{(1+x)}\]                    Þ \[f'(x)=1-\frac{4}{{{(1+x)}^{2}}}=0\]; \[{{x}^{2}}+2x-3=0\]Þ\[x=-3,\ 1\]                    Now   \[{f}''\,(x)=\frac{8}{{{(1+x)}^{3}}}\], \[{f}''\,(-3)=-ve\], \[{f}''\,(1)=+ve\]                    Hence minimum value at \[x=1\]                    \[f(1)=\frac{(5+1)(2+1)}{(1+1)}=\frac{6\times 3}{2}=9\].


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