If f(x)= x^3-x+3 and if c is the only real number such that f(c)=0, then c is between ______?

To determine the interval in which 'c' lies, we need to analyze the behavior of the function f(x) = x^3 - x + 3.

The given function is a polynomial, and for polynomials, the intermediate value theorem can be applied. According to this theorem, if a continuous function takes on positive and negative values at two points, then it must pass through zero at some point in between.

To find the interval in which 'c' lies, we will first determine the signs of f(x) for a few values of x.

Let's evaluate f(x) for x = -1, 0, and 1:
f(-1) = (-1)^3 - (-1) + 3 = -1 + 1 + 3 = 3
f(0) = (0)^3 - (0) + 3 = 3
f(1) = (1)^3 - (1) + 3 = 1 - 1 + 3 = 3

From these evaluations, we can see that the function f(x) is positive for x = -1, 0, and 1. Thus, it passes through zero somewhere between -1 and 1.

Therefore, c is between -1 and 1. The precise value of 'c' can be determined through further analysis, such as numerical methods or factoring the original equation.