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Consider the continuous function f : R → R given by f(x) = 1 / (e^x + 2e^{-x}). Statement-1: There exists some c ∈ R such that f(c) = 1/3. Statement-2: For every x ∈ R, 0 < f(x) ≤ 1/(2√2). Choose the correct option.
- Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation of Statement-1.
- Statement-1 is true, Statement-2 is false.
- Statement-1 is false, Statement-2 is true.
- Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1.
Correct answer: Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1.
Solution
Statement-1 is true because the function f(x) is continuous and takes values between 0 and 1/(2√2), which includes 1/3. Statement-2 is true as it correctly describes the range of f(x), thus supporting the existence of some c such that f(c) = 1/3.
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