Question

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 ∈ R.
Statement-2 : 0 <f(x) ≤ 1/2√2, for all x ∈ R.

(a) Statement-1 is true, statement-2 is true; statement-2 is a correct explanation of statement-1.
(b) Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for statement-1.
(c) Statement-1 is true, statement-2 is false.
(d) Statement-1 is false, statement-1 is true.

Answer 1

(a) : Using A.M.-G.M. inequality,

Using extreme-value theorem, we can say that as f is continuous, f will attain a value 1/3 at some point. Here we are able to identify the point as well.