To solve the problem, we need to analyze the quadratic equation given by:
\[
(\lambda^2 + 1)x^2 - 4\lambda x + 2 = 0
\]
We want to find the set of values of \(\lambda\) such that exactly one root of the quadratic equation lies in the interval (0, 1).
### Step 1: Identify the coefficients
In the quadratic equation \(ax^2 + bx + c = 0\):
- \(a = \lambda^2 + 1\)
- \(b = -4\lambda\)
- \(c = 2\)
### Step 2: Condition for real roots
For the quadratic to have real roots, the discriminant \(D\) must be greater than 0. The discriminant is given by:
\[
D = b^2 - 4ac
\]
Substituting the values of \(a\), \(b\), and \(c\):
\[
D = (-4\lambda)^2 - 4(\lambda^2 + 1)(2)
\]
\[
D = 16\lambda^2 - 8(\lambda^2 + 1)
\]
\[
D = 16\lambda^2 - 8\lambda^2 - 8
\]
\[
D = 8\lambda^2 - 8
\]
Setting the discriminant greater than 0:
\[
8\lambda^2 - 8 > 0
\]
\[
\lambda^2 - 1 > 0
\]
### Step 3: Solve the inequality
This can be factored as:
\[
(\lambda - 1)(\lambda + 1) > 0
\]
The critical points are \(-1\) and \(1\). Testing intervals:
- For \(\lambda < -1\), both factors are negative, product is positive.
- For \(-1 < \lambda < 1\), one factor is negative and the other is positive, product is negative.
- For \(\lambda > 1\), both factors are positive, product is positive.
Thus, the solution to the inequality is:
\[
\lambda < -1 \quad \text{or} \quad \lambda > 1
\]
### Step 4: Condition for exactly one root in (0, 1)
Next, we need to ensure that exactly one root lies in the interval (0, 1). This can be checked using the values of the function at the endpoints:
1. Calculate \(f(0)\):
\[
f(0) = 2
\]
2. Calculate \(f(1)\):
\[
f(1) = (\lambda^2 + 1)(1)^2 - 4\lambda(1) + 2 = \lambda^2 - 4\lambda + 3
\]
We require \(f(0) \cdot f(1) < 0\):
\[
2(\lambda^2 - 4\lambda + 3) < 0
\]
This simplifies to:
\[
\lambda^2 - 4\lambda + 3 < 0
\]
### Step 5: Solve the quadratic inequality
Factoring the quadratic:
\[
(\lambda - 3)(\lambda - 1) < 0
\]
The critical points are \(1\) and \(3\). Testing intervals:
- For \(\lambda < 1\), both factors are negative, product is positive.
- For \(1 < \lambda < 3\), one factor is negative and the other is positive, product is negative.
- For \(\lambda > 3\), both factors are positive, product is positive.
Thus, the solution to the inequality is:
\[
1 < \lambda < 3
\]
### Step 6: Combine the conditions
We have two conditions:
1. \(\lambda < -1\) or \(\lambda > 1\)
2. \(1 < \lambda < 3\)
The overlapping solution is:
\[
1 < \lambda < 3
\]
### Final Answer
The set of values of \(\lambda\) such that exactly one root of the quadratic equation lies in (0, 1) is:
\[
\lambda \in (1, 3)
\]
