Let f: R R be a continuous function defined by `f(x)""=1/(e^x+2e^(-x))` . Statement-1: `f(c)""=1/3,` for some `c in R` . Statement-2: `0""<""f(x)lt=1/(2sqrt(2)),` for all `x in R` . (1) Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1 (2) Statement-1 is true, Statement-2 is false (3) Statement-1 is false, Statement-2 is true (4) Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Options

A. Statement-1 is True, Statement-2 is True,Statement -2 is a correct explanation for Statement -6
B. Statement -1 True ,Statement -2 is True ,Stament -2 is not a correct explanation for Statement -!
C. Statement -1 is True Statement -2 is False
D. Statement -1 is Flase,Statement -2 is True

Answer

A

Explanation

We have `f(x)=1/(e^x+2e^(-x))`
`rArr f(x)=e^x/(e^(2x)+2)`
`rArr f'(x)=((e^(2x)+2)e^x-2e^(2x)xxe^x)/((e^(2x)+2)^2)=(e^x(-e^(2x)+2))/((e^(2x)+2)^2)`
For f(x) to attain a local maximum or minimum , we must have
`f'(x)=0 rArre^(2x)=2e^x=sqrt2`
For this value of `e^x` , we have `f(x)=1/(2sqrt2)` Clearly , `f(x)gt0`
`0ltf(x)le1/(2sqrt2) " for all "x in R`
So statement-2 is true
Since f(x) is a continuous function and every continuous function attains every value its maximum and minimum . As `0lt 1/3lt1/(2sqrt2)` . So there exits `c in R` such that
`f(c)=1/3`
Hence . both the statements are true and statement-2 is a correct explanation for statement -1