1. **State the problem:** We need to show that the function $f(x) = x^4 + x - 1$ has a real root $\alpha$ in the interval $[0.5, 1.0]$. 2. **Evaluate $f(x)$ at the endpoints:** Calculate $f(0.5)$: $$f(0.5) = (0.5)^4 + 0.5 - 1 = 0.0625 + 0.5 - 1 = -0.4375$$ Calculate $f(1.0)$: $$f(1.0) = (1.0)^4 + 1.0 - 1 = 1 + 1 - 1 = 1$$ 3. **Analyze the signs:** At $x=0.5$, $f(0.5) = -0.4375 < 0$. At $x=1.0$, $f(1.0) = 1 > 0$. 4. **Apply the Intermediate Value Theorem:** Since $f(x)$ is a polynomial, it is continuous on $[0.5, 1.0]$. Because $f(0.5) < 0$ and $f(1.0) > 0$, there must be at least one $\alpha \in [0.5, 1.0]$ such that $f(\alpha) = 0$. **Final answer:** There exists a real root $\alpha$ of $f(x) = x^4 + x - 1$ in the interval $[0.5, 1.0]$ by the Intermediate Value Theorem.