Find a function f: R → R satisfying the following conditions :
(i) f(x) is continuous on [0, ∞ )
(ii) f(x) > 0 for all x ∈ (0, ∞ )
(iii) 
=
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Sol. Let φ (x) be the integral function of function f(x) on [0, ∞ ).
φ (x) = 
Clearly, φ′ (x) = f(x) for all x ∈ [0, ∞ )
Now, 
= 
⇒ 
= 
{ φ (x)} 2
⇒ 
{ φ′ (x)} 2 = – 
{ φ (x)} 2 + 
φ (x) φ′ (x)
[Differentiating both sides with respect to x]
⇒ 
{ φ′ (x)} 2 = 
⇒ x 2 { φ′ (x)} 2 = 4x φ (x) φ′ (x) –2{ φ (x)} 2 ⇒ 
–4 
+ 2 = 0
[Dividing both sides by { φ (x)
2 }]
⇒ 
= 
⇒ x = 2 ±
⇒ 
= 
⇒ 
dx = (2 ± 
) 
dx
⇒ log φ (x) = (2 ± 
) log x + log C
⇒ φ (x) = C 
⇒ φ′( x) = (2 ± 
) C 
⇒ f(x) = (2 + 
) C 
⇒ f (x) = (2 + 
) C 
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