1. We are given the implicit equation $$x^3 + y^2 = 24$$ and asked to find the second derivative $$\frac{d^2y}{dx^2}$$ at the point $$(2,4)$$. 2. First, differentiate both sides with respect to $$x$$ to find $$\frac{dy}{dx}$$. Using implicit differentiation: $$\frac{d}{dx}(x^3) + \frac{d}{dx}(y^2) = \frac{d}{dx}(24)$$ $$3x^2 + 2y \frac{dy}{dx} = 0$$ 3. Solve for $$\frac{dy}{dx}$$: $$2y \frac{dy}{dx} = -3x^2$$ $$\frac{dy}{dx} = \frac{-3x^2}{2y}$$ 4. Next, differentiate $$\frac{dy}{dx}$$ again with respect to $$x$$ to find $$\frac{d^2y}{dx^2}$$. Use the quotient rule or implicit differentiation: $$\frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{-3x^2}{2y} \right)$$ 5. Apply the quotient rule: $$\frac{d^2y}{dx^2} = \frac{(2y)(-6x) - (-3x^2)(2 \frac{dy}{dx})}{(2y)^2}$$ 6. Simplify numerator: $$= \frac{-12xy + 6x^2 \frac{dy}{dx}}{4y^2}$$ 7. Substitute $$\frac{dy}{dx} = \frac{-3x^2}{2y}$$ into the numerator: $$-12xy + 6x^2 \left( \frac{-3x^2}{2y} \right) = -12xy - \frac{18x^4}{2y} = -12xy - \frac{9x^4}{y}$$ 8. So, $$\frac{d^2y}{dx^2} = \frac{-12xy - \frac{9x^4}{y}}{4y^2} = \frac{-12xy \cdot y - 9x^4}{4y^3} = \frac{-12xy^2 - 9x^4}{4y^3}$$ 9. Evaluate at the point $$(2,4)$$: $$x=2, y=4$$ Calculate numerator: $$-12 \cdot 2 \cdot 4^2 - 9 \cdot 2^4 = -12 \cdot 2 \cdot 16 - 9 \cdot 16 = -384 - 144 = -528$$ Calculate denominator: $$4 \cdot 4^3 = 4 \cdot 64 = 256$$ 10. Therefore, $$\frac{d^2y}{dx^2} = \frac{-528}{256} = \frac{-33}{16}$$ after simplifying by dividing numerator and denominator by 16. **Final answer:** $$\boxed{-\frac{33}{16}}$$