Ways to prove that the equation

Show at least two different ways to prove that the equation x = 2−x hasexactly one real solution. (10 points) Suppose f ∈ C[a, b], that x1 ≤ x2 . . . ≤ xn are in [a, b]. Show that thereexists a number ξ between x1 and xn with, f(ξ) = 1nXni=1f(xi).(10 points) Suppose...