First slide

Mean value theorems

Question

$If 0 < a < b < \frac{π}{b} and f (a, b) = \frac{\tan b - \tan a}{b - a}, then$

Moderate

A

$f (a, b) \geq 2$
B

$f (a, b) > 1$
C

$f (a, b) \leq 1$
D

$f (a, b) \leq - 2$

Solution

$consider the function f (x) = \tan x, defined on [a, b]$
$such that a, b \in (0, \frac{π}{2})$
$\begin{array}{l} f^{'} (c) = \frac{f (b) - f (a)}{b - a} for some c \in (a, b) \\ \Rightarrow \sec^{2} C = \frac{\tan b - \tan a}{b - a} \Rightarrow f (a, b) = \sec^{2} C \\ \Rightarrow f (a, b) > 1 [∵ \sec^{2} C > 1 as C \in (0, π / 2)] \end{array}$

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