Let f : R → R be a continuous function defined by f ( x ) = e x + 2 e − x 1 . Statement-1: f ( c ) = 3 1 , for some c ∈ R . Statement-2: 0

Question

Let f : R → R be a continuous function defined by f ( x ) = e x + 2 e − x 1 ​ . Statement-1: f ( c ) = 3 1 ​ , for some c ∈ R . Statement-2: 0 < f ( x ) ≤ 2 2 ​ 1 ​ , for all x ∈ R

TestHub · Accepted Answer

f ( x ) = e x + 2 e − x 1 ​ = e 2 x + 2 e x ​ f ′ ( x ) = ( e 2 x + 2 ) 2 ( e 2 x + 2 ) e x − 2 e 2 x ⋅ e x ​ f ′ ( x ) = 0 ⇒ e 2 x + 2 = 2 e 2 x e 2 x = 2 ⇒ e x = 2 ​ maximum f ( x ) = 4 2 ​ ​ = 2 2 ​ 1 ​ 0 < f ( x ) ≤ 2 2 ​ 1 ​ ∀ x ∈ R Since 0 < 3 1 ​ < 2 2 ​ 1 ​ ⇒ for some c ∈ R f ( c ) = 3 1 ​

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