Question 37: Given the function \(y = a{x^3} + b{x^2} + cx + d\). If the graph of the function has two extreme points, the origin and the point A(−1;−1), then the function has the equation

\(y’ = 3a{x^2} + 2bx + c\)

+ The graph of the function has the extreme point as the origin, we have

\(\left\{ \begin{array}{l}

y’\left( 0 \right) = 0\\

y\left( 0 \right) = 0

\end{array} \right. \Leftrightarrow c = d = 0\)

+ The graph of the function has a maximum point of first;−first), We have

\(\left\{ \begin{array}{l}

y’\left( { – 1} \right) = 0\\

y\left( { – 1} \right) = – 1

\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}

3a – 2b = 0\\

b – a = – 1

\end{array} \right.\)

\( \Leftrightarrow \left\{ {\begin{array}{*{20}{l}}

{a = – 2}\\

{b = – 3}

\end{array}} \right.\)

So the function is:

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