Question 49: Given two polynomial functions (y = fleft( x right), y = gleft( x right)) whose graphs are two curves in the figure. Knowing that the function graph (y = fleft( x right)) has exactly one extreme point A, the function graph (y = gleft( x right)) has exactly one extreme points are B and (AB = frac{7}{4}). How many integer values ​​of parameter m are in the interval (left( { – 5;5} right)) for the function (y = left| {left| {fleft( x right) ) – gleft( x right)} right| + m} right|) has exactly 5 extremes?

Put \(h\left( x \right) = f\left( x \right) – g\left( x \right)\), we have: \(h’\left( x \right) = f’\ left( x \right) – g’\left( x \right); h’\left( x \right) = 0 \Leftrightarrow x = {x_0}\);

\(h\left( x \right) = 0 \Leftrightarrow x = {x_1}\) or \(x = {x_2}\) (\({x_1} < {x_0} < {x_2}\));

\(h\left( {{x_0}} \right) = f\left( {{x_0}} \right) – g\left( {{x_0}} \right) = – \frac{7}{4} \).

The variation table of the function \(y = h\left( x \right)\) is:

The table of variation of the function \(y = k\left( x \right) = \left| {f\left( x \right) – g\left( x \right)} \right|\) is:

Therefore, the function \(y = k\left( x \right) + m\) also has three extremes.

Because the number of extreme points of the function \(y = \left| {k\left( x \right) + m} \right|\) is equal to the sum of the extreme points of the function \(y = k\left( x \) right) + m\) and the number of simple and odd multiples of the equation \(k\left( x \right) + m = 0\), whose function \(y = k\left( x \right) + m\) also has three extreme points, so the function \(y = \left| {\left| {f\left( x \right) – g\left( x \right)} \right| + m} \ right|\) has exactly five extremes when the equation \(k\left( x \right) + m = 0\) has exactly two simple roots (or odd multiples).

Based on the variation table of the function \(y = k\left( x \right)\), the equation \(k\left( x \right) + m = 0\) has exactly two simple roots (or odd multiples). ) if and only if \( – m \ge \frac{7}{4} \Leftrightarrow m \le – \frac{7}{4}\).

Since , \(m \le – \frac{7}{4}\) and \(m \in \left( { – 5;5} \right)\) \(m \in \left\{ { – 4; – 3; – 2} \right\}\)