If`(y^(2)-5y+3)sqrt(((1+cosx))sqrt((1+cosx)sqrt(1+cosx)...infty)) lt4` for all `x in R`, then find the all possible values of y.

Answer

To solve the inequality \((y^2 - 5y + 3) \sqrt{(1 + \cos x) \sqrt{(1 + \cos x) \sqrt{(1 + \cos x) \ldots}}} < 4\) for all \(x \in \mathbb{R}\), we will follow these steps:

### Step 1: Simplify the Infinite Nested Radical
Let:
\[ t = \sqrt{(1 + \cos x) \sqrt{(1 + \cos x) \sqrt{(1 + \cos x) \ldots}}} \]
Then, we can express \(t\) as:
\[ t = \sqrt{(1 + \cos x) t} \]
Squaring both sides gives:
\[ t^2 = (1 + \cos x) t \]
Rearranging this, we get:
\[ t^2 - (1 + \cos x) t = 0 \]
Factoring out \(t\):
\[ t(t - (1 + \cos x)) = 0 \]
Thus, \(t = 0\) or \(t = 1 + \cos x\). Since \(t\) must be non-negative, we have:
\[ t = 1 + \cos x \]

### Step 2: Determine the Range of \(t\)
The cosine function, \(\cos x\), varies between -1 and 1. Therefore:
\[ 1 + \cos x \text{ varies from } 0 \text{ to } 2 \]
Thus, \(t\) varies from \(0\) to \(2\).

### Step 3: Substitute \(t\) into the Inequality
Now, substituting \(t\) into the original inequality:
\[ (y^2 - 5y + 3)(1 + \cos x) < 4 \]
To satisfy this inequality for all \(x\), we need to consider the maximum value of \(1 + \cos x\), which is \(2\):
\[ (y^2 - 5y + 3) \cdot 2 < 4 \]
This simplifies to:
\[ y^2 - 5y + 3 < 2 \]
or:
\[ y^2 - 5y + 1 < 0 \]

### Step 4: Solve the Quadratic Inequality
Now, we need to solve the quadratic inequality:
\[ y^2 - 5y + 1 < 0 \]
First, we find the roots of the equation \(y^2 - 5y + 1 = 0\) using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{5 \pm \sqrt{25 - 4}}{2} = \frac{5 \pm \sqrt{21}}{2} \]
Let:
\[ y_1 = \frac{5 - \sqrt{21}}{2}, \quad y_2 = \frac{5 + \sqrt{21}}{2} \]

### Step 5: Determine the Intervals
The quadratic \(y^2 - 5y + 1\) opens upwards (since the coefficient of \(y^2\) is positive). Therefore, it is negative between its roots:
\[ \frac{5 - \sqrt{21}}{2} < y < \frac{5 + \sqrt{21}}{2} \]

### Conclusion
The possible values of \(y\) that satisfy the original inequality for all \(x \in \mathbb{R}\) are:
\[ y \in \left( \frac{5 - \sqrt{21}}{2}, \frac{5 + \sqrt{21}}{2} \right) \]