Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems If 2 f ( x ) + x f ( 1 x ) − 2 f ( 2 sin π ( x + 1 4 ) ) 2f(x) + x f\left(\frac{1}{x}\right) - 2f\left(\sqrt{2} \sin \pi \left(x + \frac{1}{4}\right)\right) 2 f ( x ) + x f ( x 1 ) − 2 f ( 2 sin π ( x + 4 1 ) )
Mathematics
Problem If 2 f ( x ) + x f ( 1 x ) − 2 f ( 2 sin π ( x + 1 4 ) ) 2f(x) + x f\left(\frac{1}{x}\right) - 2f\left(\sqrt{2} \sin \pi \left(x + \frac{1}{4}\right)\right) 2 f ( x ) + x f ( x 1 ) − 2 f ( 2 sin π ( x + 4 1 ) ) = 4 cos 2 ( π x 2 ) + x cos ( π x ) = 4 \cos^2\left(\frac{\pi x}{2}\right) + x \cos\left(\frac{\pi}{x}\right) = 4 cos 2 ( 2 π x ) + x cos ( x π ) ∀ x ∈ R − { 0 } \forall x \in R - \{0\} ∀ x ∈ R − { 0 } f ( 2 ) + f ( 1 2 ) = 1 f(2) + f\left(\frac{1}{2}\right) = 1 f ( 2 ) + f ( 2 1 ) = 1 f ( 2 ) + f ( 1 ) = 0 f(2) + f(1) = 0 f ( 2 ) + f ( 1 ) = 0 f ( 2 ) + f ( 1 ) = f ( 1 2 ) f(2) + f(1) = f\left(\frac{1}{2}\right) f ( 2 ) + f ( 1 ) = f ( 2 1 ) f ( 1 ) ⋅ f ( 1 2 ) ⋅ f ( 2 ) = 1 f(1) \cdot f\left(\frac{1}{2}\right) \cdot f(2) = 1 f ( 1 ) ⋅ f ( 2 1 ) ⋅ f ( 2 ) = 1 Browse the problem library