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 false statement 2 is true
B. satement 1 si true , statement 2 is true statement 2 is a correct explanation for statement 1
C. statement 1 is true statement 2 is true statement 2 is not a correct explanation for statement 2
D. statement 1 is true , statement 2 is false

Answer

2

Explanation

`f(x)=(1)/(e^(x)+2e^(-x))=(e^(x))/(e^(2x)+2)`
`therefore f(x)=((e^(2x)+2)e^(x)-2e^(2x)e^(x))/(e^(2x)+2)^(2)`
So the maximum of f(x) is
`therefore 0lt1/3lt(1)/(2(sqrt(2))`
for some c in R we get
`f(c ) =1/3`