Question 17: Knowing that the graph of the function \(y = {x^3} + 3{x^2}\) has the following form: Ask the graph of the function \(y = \left| {{x^3 } + 3{x^2}} \right|\) how many extreme points?

We have: \(y = \left| {{x^3} + 3{x^2}} \right| = \left\{ \begin{array}{l}{x^3} +3{x^ 2}\,\,\,\,\,\,{\rm{when}}\,{x^3} + 3{x^2} \ge 0 \Leftrightarrow x \ge – 3\\- {x ^3} – 3{x^2}\,\,{\rm{ when}}\,{x^3} + 3{x^2} < 0 \Leftrightarrow x < – 3\end{array} \right . = \left\{ \begin{array}{l}{x^3} + 3{x^2}\,\,\,\,\,\,{\rm{when}}\,x \ge – 3\\- {x^3} – 3{x^2}\,\,{\rm{ when}}\,x < – 3\end{array} \right.\).

So we take the symmetric part of the graph of the function \(y = {x^3} + 3{x^2}\) when x < – 3.

Based on the graph, we see that the graph of the function has 3 extreme points.

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