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Application of Derivatives question

2009 · Shift 2 · Q28
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  5. /2009 · Shift 2 · Q28

Application of Derivatives question

2009 · Shift 2 · Q28

JEE AdvancedMathematicsApplication of DerivativesMultiple correct+4 / −2
For the function f(x)=xcos⁡ 1x,x≥1,f\left( x \right) = x\cos \,{1 \over x},x \ge 1,f(x)=xcosx1​,x≥1,
  1. A
    for at least one xxx in the interval [1,∞)\left[ {1,\infty } \right)[1,∞), f(x+2)−f(x)<2f\left( {x + 2} \right) - f\left( x \right) \lt 2f(x+2)−f(x)<2
  2. B
    lim⁡x→∞f′(x)=1\mathop {\lim }\limits_{x \to \infty } f'\left( x \right) = 1x→∞lim​f′(x)=1
  3. C
    for all xxx in the interval [1,∞)f(x+2)−f(x)>2\left[ {1,\infty } \right)f\left( {x + 2} \right) - f\left( x \right) \gt 2[1,∞)f(x+2)−f(x)>2
  4. D
    f′(x)f'(x)f′(x) is strictly decreasing in the interval [1,∞)\left[ {1,\infty } \right)[1,∞)
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