Applying the IVT to show a value is attained

Since $g(x) = x^3 - 2x + 2$ is a polynomial, it is continuous on every real interval, including the closed interval $[-2, 0]$. This satisfies the hypothesis of the IVT, so we may apply the theorem on this interval.

Now evaluate the endpoints: $g(-2) = (-2)^3 - 2(-2) + 2 = -8 + 4 + 2 = -2$, and $g(0) = 0 - 0 + 2 = 2$. Since $-1$ lies strictly between $g(-2) = -2$ and $g(0) = 2$, the IVT guarantees there exists at least one $c$ in $(-2, 0)$ with $g(c) = -1$.