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CET Karnataka Engineering CET - Karnataka Engineering Solved Paper-2010

  • question_answer
    If the three function \[f(x),g(x)\] and \[h(x)\] are such that \[h(x)=f(x).g(x)\] and \[f'(x).g'(x)=c\] where c is constant, then

    A) \[h'=(x).h''(x)\]

    B) \[\frac{h(x)}{h''(x)}\]

    C) \[\frac{h''(x)}{h(x)}\]

    D) \[\frac{h(x)}{h'(x)}\]

    Correct Answer: C

    Solution :

    Given,   \[h(x)=f(x).g(x)\] and \[f'(x).g'(x)=c\] \[h''(x)=f''(x).g(x)+f'(x).g'(x)\] \[+f'(x).g'(x)+f(x).g''(x)\]                                 \[h''(x)=f''(x).g(x)+f(x).g''(x)\]                                                 \[+2f'(x).g'(x)\]                                 \[h''(x)=f''(x).g(x)+f(x).{{g}^{n}}(x)+2c\]  ….(i)                 Now, we find                                 \[\frac{f''(x)}{f(x)}+\frac{g''(x)}{g(x)}+\frac{2c}{f(x).g(x)}\]                                 \[=\frac{f''(x).g(x)+g''(x).f(x)+2c}{f(x)\,.g(x)}\]                                 \[=\frac{h''(x)}{h(x)}\]   [from Eq. (i)]


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