1. **Problem statement:** Find the candidate points for extrema and verify the necessary and sufficient conditions for extrema of the function given its derivatives. 2. **Given:** The necessary condition for extrema is that the first derivative equals zero: $$x_1 = 0 \quad \vee \quad x_2 \approx 2.12 \quad \vee \quad x_3 \approx -2.12$$ 3. **Step 1: Identify candidate points for extrema.** From the necessary condition, the candidates are: $$x_1 = 0, \quad x_2 \approx 2.12, \quad x_3 \approx -2.12$$ 4. **Step 2: Check the sufficient condition using the second derivative test.** The second derivative at $x=0$ is given as: $$f''(0) = 40 \cdot 0^3 - 90 \cdot 0 = 0$$ Since $f''(0) = 0$, the second derivative test is inconclusive at $x=0$. 5. **Step 3: Interpretation.** - The points $x_2 \approx 2.12$ and $x_3 \approx -2.12$ need to be checked similarly by evaluating $f''(x)$ at those points to determine if they are maxima, minima, or points of inflection. - At $x=0$, since $f''(0) = 0$, further analysis (such as higher derivatives or the first derivative test) is needed to classify the extremum. **Final answer:** Candidate extrema points are $x=0$, $x \approx 2.12$, and $x \approx -2.12$. The second derivative test at $x=0$ is inconclusive since $f''(0)=0$. Further analysis is required for classification.