Statement 1: Let f ( x ) be a continuous function satisfying ∫ 0 1 f ( x ) ( 4 x 2 − f ( x ) ) d x = 5 4 then the value of f ( 3 ) is equal to 15 . Statement 2: If g ( x ) is a continuous function

Question

Statement 1: Let f ( x ) be a continuous function satisfying ∫ 0 1 ​ f ( x ) ( 4 x 2 − f ( x ) ) d x = 5 4 ​ then the value of f ( 3 ) is equal to 15 . Statement 2: If g ( x ) is a continuous function on [ a , b ] and satisfying ∫ a b ​ g 2 ( x ) dx = 0 then g ( x ) = 0 .

TestHub · Accepted Answer

Given ∫ 0 1 ​ − ( f 2 ( x ) − 4 f ( x ) x 2 + 4 x 4 − 4 x 4 ) d x = 5 4 ​ ​ ∫ 0 1 ​ ( 4 x 4 − ( f ( x ) − 2 x 2 ) 2 ) d x = 5 4 ​ ​ ​ ⇒ 5 4 ​ − ∫ 0 1 ​ ( f ( x ) − 2 x 2 ) 2 d x = 5 4 ​ Hence ∫ 0 1 ​ ( f ( x ) − 2 x 2 ) 2 d x = 0 ⇒ f ( x ) = 2 x 2 ⇒ f ( 3 ) = 18 ⇒ S 1 ​ is false. Also, obviously S 2 ​ is true.

Mathematics - Definite Integration Question with Solution | TestHub

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