Question

au cours des 24 derniers mois, la valeur dune action (en $) dune compagnie a pu être modélisée par une fonction définie par parties dont la règle est \\( f(x) = \

$$\begin{cases} 3,5x + 4,5 & \\text{si } x \\in 0, 5 \\\\ 22 & \\text{si } x \\in 5, 11 \\\\ -4\\sqrt{x} - 11 + 22 & \\text{si } x \\in 11, 20 \\\\ -5|x - 23| + 25 & \\text{si } x \\in 20, 24 \\end{cases}$$

\\) détermine les 3 moments où la valeur de laction a été de 15 $. si nécessaire, arrondis ta réponse à lunité. les moments sont le \\( \boldsymbol{?} \\), \\( \boldsymbol{?} \\) et \\( \boldsymbol{?} \\) mois.

Explanation:

We need to find the values of \( x \) (in months) for which \( f(x) = 15 \) for each piece of the piece - wise function.

Step 1: Analyze the first piece \( f(x)=3.5x + 4.5\) for \(x\in[0,5]\)

We set up the equation \(3.5x + 4.5=15\)
Subtract \(4.5\) from both sides: \(3.5x=15 - 4.5=10.5\)
Divide both sides by \(3.5\): \(x=\frac{10.5}{3.5} = 3\)
Since \(3\in[0,5]\), this is a valid solution.

Step 2: Analyze the second piece \(f(x) = 22\) for \(x\in[5,11]\)

We set \(22 = 15\), but \(22
eq15\), so there is no solution in the interval \([5,11]\).

Step 3: Analyze the third piece \(f(x)=- 4\sqrt{x - 11}+22\) for \(x\in[11,20]\)

Set \(-4\sqrt{x - 11}+22 = 15\)
Subtract \(22\) from both sides: \(-4\sqrt{x - 11}=15 - 22=-7\)
Multiply both sides by \(- 1\): \(4\sqrt{x - 11}=7\)
Divide both sides by \(4\): \(\sqrt{x - 11}=\frac{7}{4}=1.75\)
Square both sides: \(x-11=(1.75)^{2}=3.0625\)
Add \(11\) to both sides: \(x=11 + 3.0625 = 14.0625\approx14\)
Since \(14\in[11,20]\), this is a valid solution.

Step 4: Analyze the fourth piece \(f(x)=-5|x - 23|+25\) for \(x\in[20,24]\)

Set \(-5|x - 23|+25 = 15\)
Subtract \(25\) from both sides: \(-5|x - 23|=15 - 25=-10\)
Divide both sides by \(-5\): \(|x - 23| = 2\)
We have two cases:

• Case 1: \(x-23=2\), then \(x=23 + 2=25\), but \(25

otin[20,24]\)

• Case 2: \(x - 23=-2\), then \(x=23-2 = 21\)

Since \(21\in[20,24]\), this is a valid solution.

Answer:

3, 14, 21