For any differentiable function f (x) on R, if x is not equal to 1 and satisfies (x-1) f '(x) > 0, it is proved that f (0) + F (2) > 2F (1)

For any differentiable function f (x) on R, if x is not equal to 1 and satisfies (x-1) f '(x) > 0, it is proved that f (0) + F (2) > 2F (1)

This problem can be explained with geometric intuition. We can construct a trapezoid in the plane rectangular coordinate system. It can be seen that f (0) and f (2) are the top and bottom of the trapezoid respectively, and dividing by 2 is the trapezoid median line. Therefore, we only need to prove that f (1) is shorter than the trapezoid median line, that is to say, f (x) is a concave function. When x is less than 1, the derivative of F (x) is less than