1. **State the problem:** We need to determine if the cubic equation $x^3 - 3x^2 - x + 4 = 0$ has a root in a given interval (though the interval is not specified, we will analyze the function behavior). 2. **Recall the Intermediate Value Theorem:** If a continuous function changes sign over an interval $[a,b]$, i.e., $f(a) \cdot f(b) < 0$, then there is at least one root in $(a,b)$. 3. **Evaluate the function at some points to check sign changes:** - At $x=0$: $f(0) = 0^3 - 3\cdot0^2 - 0 + 4 = 4$ - At $x=1$: $f(1) = 1 - 3 - 1 + 4 = 1$ - At $x=2$: $f(2) = 8 - 12 - 2 + 4 = -2$ 4. **Check sign changes:** - Between $x=1$ and $x=2$, $f(1) = 1 > 0$ and $f(2) = -2 < 0$, so the function changes sign. 5. **Conclusion:** By the Intermediate Value Theorem, there is at least one root in the interval $(1,2)$. If the question states "does not have a root in the interval" but does not specify the interval, the above shows there is a root in $(1,2)$. For other intervals, similar checks can be done. **Final answer:** The equation $x^3 - 3x^2 - x + 4 = 0$ has at least one root in the interval $(1,2)$.