Use the intermediate theorem to show that x^3+x+3=0 has a solution in interval (-2, 0)

To apply the Intermediate Value Theorem, we need to ensure that the function is continuous on the given interval and also that the function takes on both positive and negative values on that interval.

First, let's check the continuity of the function f(x) = x^3 + x + 3 on the interval (-2, 0). The function is a polynomial, and polynomials are continuous everywhere. Therefore, f(x) is continuous on (-2, 0).

Next, let's evaluate the function at the endpoints of the interval to check if it takes on both positive and negative values.

f(-2) = (-2)^3 + (-2) + 3 = -2
f(0) = (0)^3 + (0) + 3 = 3

Since f(-2) = -2 and f(0) = 3, the function f(x) takes on both positive and negative values on the interval (-2, 0).

Therefore, by the Intermediate Value Theorem, there exists a number c in the interval (-2, 0) such that f(c) = 0. This means that the equation x^3 + x + 3 = 0 has a solution in the interval (-2, 0).

Answer: The equation x^3 + x + 3 = 0 has a solution in the interval (-2, 0).