Example: Use the Intermediate Value Theorem to show that the polynomial function f(x)= x^3 + 2x - 1 has a zero in the interval (0,1).
Solution: Note that f is continuous on the closed interval (0,1). Because f(0) = 0^3+2(0) -1 = -1 and f(1) = 1^3 +2(1) -1 =2, it follows that f(0) < 0 and f(1) > 0. You can then apply the Intermediate Value Theorem to conclude that there must be some c in (0,1) such that f(c)= 0.
Source: Calculus of a Single Variable Sixth Edition book
Solution: Note that f is continuous on the closed interval (0,1). Because f(0) = 0^3+2(0) -1 = -1 and f(1) = 1^3 +2(1) -1 =2, it follows that f(0) < 0 and f(1) > 0. You can then apply the Intermediate Value Theorem to conclude that there must be some c in (0,1) such that f(c)= 0.
Source: Calculus of a Single Variable Sixth Edition book