Consider the following.
cos(x) = x³
(a) Prove that the equation has at least one real root.
The equation cos(x) = x³ is equivalent to the equation f(x) = cos(x) - x³ = 0. f(x) is continuous on the interval [0, 1], f(0) = 1, and f(1) = -0.45. Since f(1) < 0 < f(0), 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) = x³, in the interval (0, 1).
(b) Use your calculator to find an interval of length 0.01 that contains a root. (Enter your answer using interval notation. Round your answers to two decimal places.)
(0.96,0.87)