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GATE 2019 EC – Question 36

Engineering Mathematics · Calculus · 2 marks · Multiple choice

Consider a differentiable function $f(x)$ on the set of real numbers such that $f(-1)=0$ and $|f'(x)|\le2$. Given these conditions, which one of the following inequalities is necessarily true for all $x\in[-2,2]$?

  1. $f(x)\le\frac12|x+1|$
  2. $f(x)\le2|x+1|$
  3. $f(x)\le\frac12|x|$
  4. $f(x)\le2|x|$

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Correct answer: (B) $f(x)\le2|x+1|$

Explanation

By the mean value theorem, $f(x)-f(-1)=f'(c)(x+1)$ for some $c$ between $-1$ and $x$, so $|f(x)|\le2|x+1|$ and in particular $f(x)\le2|x+1|$.