1. **State the problem:** We are given the cubic function $$f(x) = 2x^3 - 9x^2 + 7x + 3$$ and need to find its zeros (roots) and any local maximum or minimum values. 2. **Find the zeros:** To find zeros, solve $$f(x) = 0$$: $$2x^3 - 9x^2 + 7x + 3 = 0$$ 3. **Use the Rational Root Theorem and synthetic division or numerical methods:** Testing possible rational roots, we find one root approximately at $$x \approx -0.333$$. Using polynomial division, factor out $$x + 0.333$$ and solve the quadratic remainder: $$2x^3 - 9x^2 + 7x + 3 = (x + 0.333)(2x^2 - 9.666x + 9)$$ 4. **Solve the quadratic:** Use the quadratic formula: $$x = \frac{9.666 \pm \sqrt{9.666^2 - 4 \cdot 2 \cdot 9}}{2 \cdot 2}$$ Calculate discriminant: $$\Delta = 9.666^2 - 72 = 93.44 - 72 = 21.44$$ Square root: $$\sqrt{21.44} \approx 4.63$$ So, $$x = \frac{9.666 \pm 4.63}{4}$$ Two roots: $$x_1 = \frac{9.666 + 4.63}{4} = \frac{14.296}{4} = 3.574$$ $$x_2 = \frac{9.666 - 4.63}{4} = \frac{5.036}{4} = 1.259$$ 5. **Zeros in numerical order:** $$x \approx -0.333, 1.259, 3.574$$ 6. **Find local maxima and minima:** Take the derivative: $$f'(x) = 6x^2 - 18x + 7$$ Set derivative to zero to find critical points: $$6x^2 - 18x + 7 = 0$$ Use quadratic formula: $$x = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 6 \cdot 7}}{2 \cdot 6} = \frac{18 \pm \sqrt{324 - 168}}{12} = \frac{18 \pm \sqrt{156}}{12}$$ $$\sqrt{156} \approx 12.49$$ Critical points: $$x_1 = \frac{18 - 12.49}{12} = \frac{5.51}{12} = 0.459$$ $$x_2 = \frac{18 + 12.49}{12} = \frac{30.49}{12} = 2.541$$ 7. **Determine nature of critical points using second derivative:** $$f''(x) = 12x - 18$$ Evaluate at $$x_1 = 0.459$$: $$f''(0.459) = 12(0.459) - 18 = 5.508 - 18 = -12.492 < 0$$ so local maximum. Evaluate at $$x_2 = 2.541$$: $$f''(2.541) = 12(2.541) - 18 = 30.492 - 18 = 12.492 > 0$$ so local minimum. 8. **Find function values at critical points:** $$f(0.459) = 2(0.459)^3 - 9(0.459)^2 + 7(0.459) + 3 \approx 3.637$$ $$f(2.541) = 2(2.541)^3 - 9(2.541)^2 + 7(2.541) + 3 \approx -4.637$$ **Final answers:** - Zeros: $$x \approx -0.333, 1.259, 3.574$$ - Local maximum at $$x \approx 0.459$$ with $$f(x) \approx 3.637$$ - Local minimum at $$x \approx 2.541$$ with $$f(x) \approx -4.637$$