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Question:

Let f:R→R be a continuous function defined by f(x) = 1/(ex + 2e-x). Statement-1: f(c) = 1/3, for some c∈R. Statement-2: 0 < f(x) ≤ 1/(2√2), for all x∈R. Choose the correct option regarding the truth values of Statement-1 and Statement-2.

Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1

Statement-1 is false, Statement-2 is true

Statement-1 is true, Statement-2 is false

Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1

Solution:

f(x) = 1/(ex + 2e-x) = ex/(e2x + 2) ⇒ f'(x) = (e2x + 2)ex - ex(2e2x)/(e2x + 2)2
For critical point f'(x) = 0 ⇒ ex = √2/2
Maximum value f(x) = √2/4 = 1/(2√2)
0 < f(x) ≤ 1/(2√2) ∀x ∈ R
Since 0 < 1/3 < 1/(2√2) = for some c ∈ R ⇒ f(c) = 1/3
Hence, option 'A' is correct.