1. The problem is to analyze the function $$f(x) = \frac{1}{3}x^{\frac{4}{3}} - x^{\frac{1}{3}}$$ for $$x \in [0,3]$$. 2. This function involves fractional exponents. Recall that $$x^{\frac{m}{n}} = \sqrt[n]{x^m}$$. 3. We will first simplify and understand the function: $$f(x) = \frac{1}{3}x^{\frac{4}{3}} - x^{\frac{1}{3}}$$ 4. To find critical points or analyze behavior, we might consider the derivative, but since the problem only states the function and domain, we focus on understanding the function on $$[0,3]$$. 5. Evaluate the function at the endpoints: At $$x=0$$: $$f(0) = \frac{1}{3} \cdot 0^{\frac{4}{3}} - 0^{\frac{1}{3}} = 0 - 0 = 0$$ At $$x=3$$: Calculate $$3^{\frac{1}{3}} = \sqrt[3]{3}$$ and $$3^{\frac{4}{3}} = 3^{1 + \frac{1}{3}} = 3 \cdot \sqrt[3]{3}$$. So, $$f(3) = \frac{1}{3} \cdot 3 \cdot \sqrt[3]{3} - \sqrt[3]{3} = \cancel{\frac{1}{3} \cdot 3} \cdot \sqrt[3]{3} - \sqrt[3]{3} = \sqrt[3]{3} - \sqrt[3]{3} = 0$$ 6. The function equals zero at both ends of the interval. 7. To understand the shape, consider the derivative: $$f'(x) = \frac{1}{3} \cdot \frac{4}{3} x^{\frac{4}{3} - 1} - \frac{1}{3} x^{\frac{1}{3} - 1} = \frac{4}{9} x^{\frac{1}{3}} - \frac{1}{3} x^{-\frac{2}{3}}$$ 8. Set derivative to zero to find critical points: $$\frac{4}{9} x^{\frac{1}{3}} = \frac{1}{3} x^{-\frac{2}{3}}$$ Multiply both sides by $$x^{\frac{2}{3}}$$: $$\frac{4}{9} x^{\frac{1}{3} + \frac{2}{3}} = \frac{1}{3}$$ $$\frac{4}{9} x^{1} = \frac{1}{3}$$ $$x = \frac{1/3}{4/9} = \frac{1}{3} \times \frac{9}{4} = \frac{3}{4} = 0.75$$ 9. So, the critical point is at $$x=0.75$$. 10. Evaluate $$f(0.75)$$: Calculate $$0.75^{\frac{1}{3}}$$ and $$0.75^{\frac{4}{3}} = (0.75^{\frac{1}{3}})^4$$. Approximate $$0.75^{\frac{1}{3}} \approx 0.908$$. Then, $$0.75^{\frac{4}{3}} = 0.908^4 \approx 0.681$$. So, $$f(0.75) = \frac{1}{3} \times 0.681 - 0.908 = 0.227 - 0.908 = -0.681$$ 11. The function dips to approximately $$-0.681$$ at $$x=0.75$$ and returns to zero at $$x=3$$. Final answer: The function $$f(x) = \frac{1}{3}x^{\frac{4}{3}} - x^{\frac{1}{3}}$$ on $$[0,3]$$ starts at 0, decreases to about $$-0.681$$ at $$x=0.75$$, and returns to 0 at $$x=3$$.