1. **Problem Statement:** Determine if the function $f(x) = x^3 - 5x^2 + 5x + 1$ is continuous at $x = \sqrt{2}$. 2. **Recall the definition of continuity at a point:** A function $f$ is continuous at $x = c$ if three conditions hold: 1. $f(c)$ is defined. 2. The limit $\lim_{x \to c} f(x)$ exists. 3. $\lim_{x \to c} f(x) = f(c)$. 3. **Check if $f(\sqrt{2})$ is defined:** Since $f$ is a polynomial, it is defined for all real numbers, so $f(\sqrt{2})$ exists. 4. **Evaluate $f(\sqrt{2})$:** $$f(\sqrt{2}) = (\sqrt{2})^3 - 5(\sqrt{2})^2 + 5(\sqrt{2}) + 1 = 2\sqrt{2} - 5 \times 2 + 5\sqrt{2} + 1 = 2\sqrt{2} - 10 + 5\sqrt{2} + 1$$ Combine like terms: $$2\sqrt{2} + 5\sqrt{2} = 7\sqrt{2}$$ $$-10 + 1 = -9$$ So, $$f(\sqrt{2}) = 7\sqrt{2} - 9$$ 5. **Check the limit $\lim_{x \to \sqrt{2}} f(x)$:** Since $f$ is a polynomial, it is continuous everywhere, so the limit equals the function value at that point. 6. **Conclusion:** Since $f(\sqrt{2})$ is defined and the limit equals the function value, $f$ is continuous at $x = \sqrt{2}$. **Final answer:** $f$ is continuous at $x = \sqrt{2}$.