1. **State the problem:** Given the function $y=2x^3 + 7x^2 + 4x - 3$, we need to find the stationary points, the point of inflection, and sketch the curve. 2. **Recall formulas and rules:** - Stationary points occur where the first derivative $y'$ equals zero. - Points of inflection occur where the second derivative $y''$ equals zero and the concavity changes. 3. **Find the first derivative:** $$y' = \frac{dy}{dx} = 6x^2 + 14x + 4$$ 4. **Find stationary points by solving $y' = 0$:** $$6x^2 + 14x + 4 = 0$$ Divide entire equation by 2 for simplicity: $$3x^2 + 7x + 2 = 0$$ Factor or use quadratic formula: $$x = \frac{-7 \pm \sqrt{7^2 - 4 \times 3 \times 2}}{2 \times 3} = \frac{-7 \pm \sqrt{49 - 24}}{6} = \frac{-7 \pm 5}{6}$$ So, - $$x = \frac{-7 + 5}{6} = \frac{-2}{6} = -\frac{1}{3}$$ - $$x = \frac{-7 - 5}{6} = \frac{-12}{6} = -2$$ 5. **Find corresponding $y$ values:** - For $x = -\frac{1}{3}$: $$y = 2\left(-\frac{1}{3}\right)^3 + 7\left(-\frac{1}{3}\right)^2 + 4\left(-\frac{1}{3}\right) - 3 = 2\left(-\frac{1}{27}\right) + 7\left(\frac{1}{9}\right) - \frac{4}{3} - 3 = -\frac{2}{27} + \frac{7}{9} - \frac{4}{3} - 3$$ Convert to common denominator 27: $$-\frac{2}{27} + \frac{21}{27} - \frac{36}{27} - \frac{81}{27} = \frac{-2 + 21 - 36 - 81}{27} = \frac{-98}{27}$$ - For $x = -2$: $$y = 2(-2)^3 + 7(-2)^2 + 4(-2) - 3 = 2(-8) + 7(4) - 8 - 3 = -16 + 28 - 8 - 3 = 1$$ 6. **Find the second derivative:** $$y'' = \frac{d^2y}{dx^2} = 12x + 14$$ 7. **Find points of inflection by solving $y'' = 0$:** $$12x + 14 = 0 \Rightarrow x = -\frac{14}{12} = -\frac{7}{6}$$ 8. **Find corresponding $y$ value at point of inflection:** $$y = 2\left(-\frac{7}{6}\right)^3 + 7\left(-\frac{7}{6}\right)^2 + 4\left(-\frac{7}{6}\right) - 3$$ Calculate stepwise: $$2 \times \left(-\frac{343}{216}\right) + 7 \times \frac{49}{36} - \frac{28}{6} - 3 = -\frac{686}{216} + \frac{343}{36} - \frac{28}{6} - 3$$ Convert all to denominator 216: $$-\frac{686}{216} + \frac{2058}{216} - \frac{1008}{216} - \frac{648}{216} = \frac{-686 + 2058 - 1008 - 648}{216} = \frac{-284}{216} = -\frac{71}{54}$$ 9. **Summary:** - Stationary points at $\left(-2, 1\right)$ and $\left(-\frac{1}{3}, -\frac{98}{27}\right)$. - Point of inflection at $\left(-\frac{7}{6}, -\frac{71}{54}\right)$. 10. **Sketching the curve:** - The curve has two stationary points: one local maximum or minimum at $x=-2$ and another at $x=-\frac{1}{3}$. - The point of inflection at $x=-\frac{7}{6}$ indicates where the curve changes concavity. This completes the analysis of the curve.