(50) Solve carefully plz

Let (X1, ..., Xn) be a random sample of random variables with Lebesgue density f(x), where f is a Lebesgue density on (0,) or symmetric about 0, and > 0 is an unknown parameter. Show that the likelihood equation has a unique root if xf (x)/f(x) is continuous and decreasing for x > 0. Verify that this condition is satisfied if f(x) = 1(1 + x2)1.