1. **State the problem:** We have a cubic function $f(x) = a x^3 - 2 x^2 + 4$ and know it has an inflection point at $x = \frac{1}{3}$. We need to find the value of $\frac{f(1)}{f(1)}$. 2. **Recall the definition of an inflection point:** An inflection point occurs where the second derivative $f''(x)$ changes sign, which means $f''(x) = 0$ at that point. 3. **Find the first derivative:** $$f'(x) = \frac{d}{dx}(a x^3 - 2 x^2 + 4) = 3 a x^2 - 4 x$$ 4. **Find the second derivative:** $$f''(x) = \frac{d}{dx}(3 a x^2 - 4 x) = 6 a x - 4$$ 5. **Use the inflection point condition:** Set $f''\left(\frac{1}{3}\right) = 0$: $$6 a \cdot \frac{1}{3} - 4 = 0 \implies 2 a - 4 = 0 \implies 2 a = 4 \implies a = 2$$ 6. **Substitute $a=2$ back into $f(x)$:** $$f(x) = 2 x^3 - 2 x^2 + 4$$ 7. **Calculate $f(1)$:** $$f(1) = 2 \cdot 1^3 - 2 \cdot 1^2 + 4 = 2 - 2 + 4 = 4$$ 8. **Calculate $\frac{f(1)}{f(1)}$:** $$\frac{f(1)}{f(1)} = \frac{4}{4} = 1$$ **Final answer:** $\boxed{1}$ which corresponds to option (c).