Consider the following: cos(x) = x^3
The equation cos(x) = x^3 is equivalent to the equation f(x) = cos(x) - x^3 = 0. f(x) is continuous on the interval [0, 1], f(0) = 1 and f(1) = Since there is a number c in (0, 1) such that f(c) = 0 by the Intermediate Value Theorem. Thus, there is a root of the equation cos(x) = x^3, n the interval (0, 1).
(a) Prove that the equation has at least one real root.
(b) Use a calculator to find an interval of length 0.01 that contains a solution.