1. The problem asks us to find the intervals where the function $f(x) = x^4 - 24x^2 + 4$ is convex downward. Convex downward means the graph is concave down, which occurs where the second derivative $f''(x) < 0$. 2. First, find the first derivative: $$f'(x) = \frac{d}{dx}(x^4 - 24x^2 + 4) = 4x^3 - 48x$$ 3. Next, find the second derivative: $$f''(x) = \frac{d}{dx}(4x^3 - 48x) = 12x^2 - 48$$ 4. To find where the function is concave down, solve for when: $$f''(x) < 0$$ $$12x^2 - 48 < 0$$ 5. Simplify the inequality: $$12x^2 < 48$$ $$x^2 < 4$$ 6. Taking the square root on both sides gives: $$-2 < x < 2$$ 7. This corresponds to the open interval $] -2, 2[$. 8. Checking the graph and sign analysis in the question confirms that $f''(x)$ is negative between $-2$ and $2$, meaning the graph is concave downward there. Final answer: The curve is convex downward on the interval $] -2, 2[$ which corresponds to option C).