Mean Value Theorem

Question

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?

Comments

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

You can post it as the answer instead of a comment.

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@ techbd123 Done :)

Answer 1

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.