Question 15: The graph of the function \(y = – {x^3} + 3{x^2} + 9x + 1\) has two extreme points A and B. Which of the following points is on the line AB ?

Determination set

\(y’ = – 3{x^2} + 6x + 9\)

\(y’ = 0 \Leftrightarrow – 3{x^2} + 6x + 9 = 0 \Leftrightarrow \left[\begin{array}{l}x=–1\\x=3\\end{array}\right\)[\begin{array}{l}x=–1\x=3\\end{array}\right\)

Therefore, the graph of the function has two extreme points, \(A\left( { – 1; – 4} \right)\) and \(B\left( {3;28} \right)\)

Infer the line AB with the equation \(8x – y + 4 = 0\)

Substituting \(N\left( {1;\,12} \right)\) into equation AB we get 8.1 – 12 + 4 = 0.

So N belongs to AB.

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