Let f x = 9 x 25 + c , c > 0 . If the curve y = f - 1 x passes through 1 4 , - 5 4 and g x is the antiderivative of f - 1 x such that g 0 = 5 2 , then the value of g 1 is, (where [ . ] represents the greatest integer function)

Step-by-step solution

y = f x = 9 x 25 + c 1 ⇒ x = 25 9 y - c 1 = f - 1 y ⇒ f - 1 x = 25 9 x - c 1 The curve y = f - 1 x passes through 1 4 , - 5 9 ⇒ - 5 9 = 25 9 1 4 - c 1 ⇒ c 1 = 1 4 + 1 5 = 9 20 ⇒ f - 1 x = 25 9 x - 9 20 = 25 x 9 - 5 4 g x = ∫ f - 1 x d x = ‍ ∫ 25 x 9 - 5 4 d x ‍ = 25 9 x 2 2 - 5 x 4 + c 2 g 0 = 5 2 ⇒ c 2 = 5 2 ⇒ g x = 25 9 x 2 2 - 5 x 4 + 5 2 ⇒ g 1 = 25 18 - 5 4 + 5 2 = 50 - 45 + 90 36 = 95 36 ⇒ g 1 = 2