Let f be a continuous function and satisfies $f^{'} (x) > 0$ on $(0, \infty)$ and the value of $f^{''} (x), \forall x \in (0, \infty)$ is always equal to minimum value of the function

Question

Let f be a continuous function and satisfies f ′ ( x ) > 0 on ( 0 , ∞ ) and the value of f ′′ ( x ) , ∀ x ∈ ( 0 , ∞ ) is always equal to minimum value of the function

TestHub · Accepted Answer

​ ⇒ x ∈ R min ​ { e − ∣ x ∣ + 2 , ∣ x ∣ + 2 } = 2 ⇒ f ′′ ( x ) = 2 , ∀ x ∈ ( 0 , ∞ ) L = x → ∞ lim ​ f ′ ( x ) ( 6 x + ( x 2 + 1 ) 2 6 x ​ − 4 f ′′ ( x ) ) ​ (By L’H o ˆ pital’s Rule) L = x → ∞ lim ​ ​ f ′′ ( x ) 6 − 6 ( 1 + x 2 ) 3 ( 1 − 3 x 2 ) ​ − 0 ​ ​ = 2 6 − 0 ​ = 3 ⇒ L = 3 ​

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