1. The problem is to find the third derivative $f^{(3)}(x)$ of the function $f(x) = 3e^{-2x} + 5x^{4}$. 2. Recall the rules for derivatives: - The derivative of $e^{ax}$ is $ae^{ax}$. - The derivative of $x^n$ is $nx^{n-1}$. 3. First, find the first derivative: $$f'(x) = 3 \cdot (-2)e^{-2x} + 5 \cdot 4x^{3} = -6e^{-2x} + 20x^{3}$$ 4. Next, find the second derivative: $$f''(x) = -6 \cdot (-2)e^{-2x} + 20 \cdot 3x^{2} = 12e^{-2x} + 60x^{2}$$ 5. Finally, find the third derivative: $$f^{(3)}(x) = 12 \cdot (-2)e^{-2x} + 60 \cdot 2x = -24e^{-2x} + 120x$$ 6. Therefore, the correct third derivative is: $$f^{(3)}(x) = -24e^{-2x} + 120x$$ This matches the expression given in the top-left option.