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

  • question_answer
    Let \[h(x)=f(x)-{{(f(x))}^{2}}+{{(f(x))}^{3}}\] for every real number x. Then  [IIT 1998]

    A) h is increasing whenever f is increasing

    B) h is increasing whenever f  is decreasing

    C) h is decreasing whenever f is decreasing

    D) Nothing can be said in general

    Correct Answer: A

    Solution :

    • \[h(x)=f(x)-{{[f(x)]}^{2}}+{{[f(x)]}^{3}}\]                   
    • \[h'(x)=f'(x)-2f(x)f'(x)+3{{[f(x)]}^{2}}f'(x)\]                           
    • \[=f'(x)[1-2f(x)+3{{[f(x)]}^{2}}]\]                           
    • \[=3f'(x)\left\{ {{\left( f(x)-\frac{1}{3} \right)}^{2}}+\frac{2}{9} \right\}\]                   
    • \[\therefore \]\[h'(x)\]and \[f'(x)\]have same sign.


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