Example 1. Investigate the convexity, concavity, and points of inflection of the function $y=x^{3}+3 x^{2}+6 x+7$.
Solution. We have: \( y' = 3x^2 + 6x + 6, y'' = 6x + 6 \). The signs of \( y'' \) are indicated on the number line.

Answer. In the interval \( (-\infty, -1) \), \( f(x) \) is convex, in the interval \( (-1, +\infty) \), \( f(x) \) is concave, \( x = -1 \) is the abscissa of the inflection point (Fig. 7.10) and \( y_{\text{inf}} = y(-1) = 3 \).

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-089.jpg?height=209&width=424&top_left_y=336&top_left_x=147)

Fig. 7.10

![](https://cdn.mathpix.com/cropped/2024_05_22_8f5be56c3f01f571fd60g-089.jpg?height=213&width=447&top_left_y=331&top_left_x=704)

Fig. 7.11