1. **State the problem:** We are given the function $F(x) = 3x^4 - 2x^2 + 7$ and we want to analyze or work with it. 2. **Identify the type of function:** This is a polynomial function of degree 4. 3. **Important rules:** Polynomial functions are continuous and differentiable everywhere. The degree and leading coefficient determine the end behavior. 4. **Find the derivative:** To find critical points or analyze the function's behavior, compute the derivative: $$F'(x) = \frac{d}{dx}(3x^4 - 2x^2 + 7) = 12x^3 - 4x$$ 5. **Set derivative to zero to find critical points:** $$12x^3 - 4x = 0$$ Factor out $4x$: $$4x(3x^2 - 1) = 0$$ So, $$4x = 0 \Rightarrow x = 0$$ $$3x^2 - 1 = 0 \Rightarrow x^2 = \frac{1}{3} \Rightarrow x = \pm \frac{1}{\sqrt{3}}$$ 6. **Summary:** The critical points are at $x = 0$, $x = \frac{1}{\sqrt{3}}$, and $x = -\frac{1}{\sqrt{3}}$. This completes the analysis of the function's critical points. **Final answer:** $$F'(x) = 12x^3 - 4x$$ Critical points at $x = 0$, $x = \pm \frac{1}{\sqrt{3}}$.