An animation showing an inscribed rectangle under the curve y = 4 - x^2, symmetric about the y-axis. It builds the area function A(x) = 2x(4 - x^2), differentiates to find dA/dx = 8 - 6x^2, solves for the critical point x = 2/sqrt(3), and reveals the maximum area 32sqrt(3)/9. Useful for calculus students learning optimization problems involving geometric constraints and derivative-based maxima.
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Create a 30-second Manim-style math animation. Problem: For the parabola y = 4 - x^2, a rectangle is symmetric about the y-axis with its base on the x-axis and upper corners on the parabola. Show only these steps: 1. Draw y = 4 - x^2. 2. Draw the rectangle. 3. Label half-width = x. 4. Height = 4 - x^2. 5. Area = 2x(4 - x^2). 6. Differentiate: dA/dx = 8 - 6x^2. 7. Set dA/dx = 0. 8. Show x = 2/sqrt(3). 9. Show maximum area = 32sqrt(3)/9. Use clean mathematical animation. No narration. No extra explanation. Do not change any equation.