1. **State the problem:** We are given the function $f(x) = 9x^6 - 9 + 4x^4$ and want to understand it better. 2. **Rewrite the function in standard polynomial form:** $$f(x) = 9x^6 + 4x^4 - 9$$ 3. **Look for important features:** - The function is a polynomial of degree 6. - The leading term $9x^6$ dominates for large $|x|$, so as $x \to \pm \infty$, $f(x) \to +\infty$. 4. **Find critical points by differentiating:** $$f'(x) = \frac{d}{dx}(9x^6 + 4x^4 - 9) = 54x^5 + 16x^3$$ 5. **Set derivative equal to zero to find critical points:** $$54x^5 + 16x^3 = 0$$ Factor out $x^3$: $$x^3(54x^2 + 16) = 0$$ 6. **Solve for $x$:** - $x^3 = 0 \implies x = 0$ - $54x^2 + 16 = 0 \implies x^2 = -\frac{16}{54}$ which has no real solutions. 7. **Conclusion on critical points:** Only one critical point at $x=0$. 8. **Evaluate $f(0)$:** $$f(0) = 9(0)^6 + 4(0)^4 - 9 = -9$$ 9. **Interpretation:** - The function has a minimum or maximum at $x=0$. - Since $f(x) \to +\infty$ as $x \to \pm \infty$ and $f(0) = -9$, this point is a local minimum. 10. **Summary:** - The function $f(x) = 9x^6 + 4x^4 - 9$ is a polynomial with a single local minimum at $x=0$ where $f(0) = -9$. - For large $x$, $f(x)$ grows very large positively. This explains the behavior and shape of the function clearly.