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ISI2015-MMA-74

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Let $f$ and $g$ be two differentiable functions such that $f’(x)\leq g’(x)$for all $x<1$ and $f’(x) \geq g’(x)$ for all $x>1$. Then

  1. if $f(1) \geq g(1)$, then $f(x) \geq g(x)$ for all $x$
  2. if $f(1) \leq g(1)$, then $f(x) \leq g(x)$ for all $x$
  3. $f(1) \leq g(1)$
  4. $f(1) \geq g(1)$
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Answer: C

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Let $f(x)=\frac{x^2}{2}$ and $g(x)=x$ which statisfies the hypothesis of the question.

 

This gives $f(1)\le g(1)$

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