text: # Mean Value Theorem
  
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  f(x) is a differentiable function that satisfies 5 ≤ f′(x) ≤ 14 for all x. Let a and b be the maximum and minimum values, respectively, that f(11)−f(3) can possibly have, then what is the value of a+b?
  in Calculus
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  can possibly have?
  
  What?
  
  Is the question complete?
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  Yes the question is complete. You can check it here:
  
  https://brilliant.org/wiki/mean-value-theorem/
  
  1
  
  Solution:
  
  Since f(x) is differentiable on all intervals, we can choose any two points (Here we take 11 and 3). So from the mean value theorem, we have some c such that
  
  $f'(c) = (f(11)-f(3))/11-3$)
  
  Now given  $5 \leqslant f'(c)\leqslant14$  therefore   $5 \leqslant (f(11)-f(3))/8 \leqslant14$
  
  =>  $40 \leqslant (f(11)-f(3)) \leqslant112$
  
  Hence a = 112 and b = 40  which gives a+b = 152 as the answer.
  
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  @Nirmal Gaur
  
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  Done :)
  
  1 vote
  Since $f(x)$ is differentiable on all intervals, we can choose any two points $($ Here we take $11$ and $3)$. So from the mean value theorem, we have some $c$ such that:
  
  $f'\left ( c \right )=\dfrac{\bigg(f(11)-f(3)\bigg)}{(11-3)}$
  
  Now given $5\leqslant f'\left ( c \right )\leqslant 14$  therefore:
  
  $\implies 5\leqslant \dfrac{\bigg(f(11)-f(3)\bigg)}{8}\leqslant 14$
  
  $\implies 40\leqslant \bigg(f(11)-f(3)\bigg)\leqslant 112$
  
  Hence $a = 112$ and $b = 40$  which gives $a+b = 152$ as the answer.
  
  edited
  
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