Prove that f(x) = x^3-sin(pi(x))/x has atleast one real zero in the interval [1/2,1]

Answer 1

Note that f is continuous on the closed interval [1/2, 1] with f(1/2) = 1/8 - 2 < 0 and f(1) = 1 > 0. By the intermediate value theorem, one concludes that there exists a zero of f in the interval [1/2, 1].