If `0lt a lt b lt(pi)/(2)` and `f(a,b)=(tan b-tana)/(b-a)`, then

Options

A. `f(a,b)ge2`
B. `f(a,b)gt2`
C. `f(a,b)le2`
D. none of these

Answer

D

Explanation

Consider the function f(x) = tan x defined on [a,b] such that `0 le a lt b lt (pi)/(2)` .
Applying Largrange's mean valeu theorem, we have
`f'(c)=(f(b)-f(a))/(b-a)"for some"c in (a,b)`
`rArr sec^(2)c=(tan b- tan a)/(b-a)`
`rArr f(a,b)= sec^(2)c`
`rArr f(a,b) gt1" " [ :. sec^(2) c gt "as" c in (0, pi//2)]`