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

  • question_answer
    If \[f(x+y)=f(x).f(y)\]for all x and y and \[f(5)=2\], \[f'(0)=3\], then \[f'(5)\]will be [IIT 1981; Karnataka CET 2000; UPSEAT 2002; MP PET 2002; AIEEE 2002]

    A) 2

    B) 4

    C) 6

    D) 8

    Correct Answer: C

    Solution :

    • Let \[x=5,\,\,\,y=0\Rightarrow f(5+0)=f(5).f(0)\]                   
    • Þ \[f(5)=f(5)f(0)\Rightarrow f(0)=1\]                   
    • Therefore, \[f'(5)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(5+h)-f(5)}{h}\]                   
    • \[=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(5)f(h)-f(5)}{h}=\underset{h\to 0}{\mathop{\lim 2}}\,\left[ \frac{f(h)-1}{h} \right]\],  \[\left\{ \because f(5)=2 \right\}\]                   
    • \[=2\underset{h\to 0}{\mathop{\lim }}\,.\left[ \frac{f(h)-f(0)}{h} \right]=2\times f'(0)=2\times 3=6\].


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