1. **State the problem:** We are given the function $f(x) = 3x^4 - 5x^2 + 7$ and we want to understand its behavior or solve related questions. 2. **Identify the function type:** This is a polynomial function of degree 4. 3. **Find the derivative to analyze critical points:** $$f'(x) = \frac{d}{dx}(3x^4 - 5x^2 + 7) = 12x^3 - 10x$$ 4. **Set the derivative equal to zero to find critical points:** $$12x^3 - 10x = 0$$ 5. **Factor the equation:** $$2x(6x^2 - 5) = 0$$ 6. **Solve for $x$:** - $2x = 0 \Rightarrow x = 0$ - $6x^2 - 5 = 0 \Rightarrow 6x^2 = 5 \Rightarrow x^2 = \frac{5}{6} \Rightarrow x = \pm \sqrt{\frac{5}{6}}$ 7. **Summary:** The critical points are at $x = 0$, $x = \sqrt{\frac{5}{6}}$, and $x = -\sqrt{\frac{5}{6}}$. These points can be used to analyze the function's maxima, minima, or points of inflection.