Question 1: Let S be the set of all integer values of parameter m so that the minimum point of the graph of the function y = x3+ x2+ mx – 1 lies to the right of the vertical axis. Find the number of integer elements of the set (−5;6)∩S.

We have the derivative y’ = 3x^{2}+ 2x+ m.

The function has an extreme when

Since the function has \( \Rightarrow {x_{CT}} > {x_{CD}}\)

Require the problem to become an equation y’ = 0 with at least 1 positive solution

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

{x_1} + {x_2} = – \frac{2}{3} < 0\\

{x_1}{x_2}\; = \frac{m}{3}

\end{array} \right. \Rightarrow m < 0\;\) is the value to look for.

So \(\left( { – 5;6} \right) \cap S = \left( { – 5;0} \right)\).

For which m integer should choose -4; -3; -2; -first. There are 4 satisfying values.

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