Let f ( x ) be a positive differentiable function on [ 0 , a ] such that f ( o ) = 1 and f ( a ) = 3 1/4 If f ′ ( x ) ≥ ( f ( x ) ) 3 + ( f ( x ) ) − 1 s then, maximum value of a is

Question

Let f ( x ) be a positive differentiable function on [ 0 , a ] such that f ( o ) = 1 and f ( a ) = 3 1/4 If f ′ ( x ) ≥ ( f ( x ) ) 3 + ( f ( x ) ) − 1 s ​ then, maximum value of a is

TestHub · Accepted Answer

f ′ ( x ) f ( x ) ≥ ( f ( x ) ) 4 + 1 ​ ⇒ { ( f ( x ) ) 2 } 2 + 1 2 f ′ ( x ) f ( x ) ​ ≥ 2 ⇒ ∫ 0 a ​ { ( f ( x ) ) 2 } 2 + 1 2 f ′ ( x ) f ( x ) ​ ≥ 2 ∫ 0 a ​ 1 d x ⇒ ​ tan − 1 ( f ( x ) ) 2 ​ 0 a ​ ≥ 2 a ⇒ 3 π ​ − 4 π ​ ≥ 2 a ​

Mathematics - Application of Derivative Question with Solution | TestHub

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