The number of solutions of the equation | ( x)| = 1 50 (x^2 + 1) in R is

Step-by-step solution

Let f(x) = | ( x)| and g(x) = 1 50 (x^2 + 1) . Both f(x) and g(x) are even functions, so their graphs are symmetric about the y-axis. We will first find the number of solutions for x > 0 . At x = 0 , f(0) = 0 and g(0) = 1 50 , so x = 0 is not a solution. Since the maximum value of f(x) is 1 , any solution must satisfy g(x) 1 . 1 50 (x^2 + 1) 1 x^2 + 1 50 x^2 49 -7 x 7 Thus, all positive solutions must lie in the interval (0, 7) . We can divide this into 7 sub-intervals: (0, 1), (1, 2), , (6, 7) . In each interval (