From the Intermediate Value Theorem, we can conclude that \( f(x)=3 x^{3}-3 x^{2}-9 x+6 \) has a zero over the interval \( [0,1] \). True False

To apply the Intermediate Value Theorem (IVT), we first need to evaluate \( f(x) \) at the endpoints of the interval. Calculating, we find \( f(0) = 6 \) and \( f(1) = -3 \). Since \( f(0) > 0 \) and \( f(1) < 0 \), the function changes sign over the interval \([0,1]\). According to the IVT, there must be at least one zero in that interval. So the statement is **True**! And just to keep the fun rolling, did you know that the Intermediate Value Theorem essentially states that for any value between \(f(a)\) and \(f(b)\), there exists some \(c\) in \((a, b)\) where \(f(c)\) equals that value? It's like a mathematical guarantee for continuity lovers! Also, if you're curious about how this theorem plays out in the real world, think about how it applies in physics and engineering when dealing with situations like finding the equilibrium point of a system—once two states of a system lead to different outcomes, you can be sure there's a solution somewhere pulling the strings in between!