To find the number of integral values of \( \lambda \) for which the inequality \( x^2 - 2\lambda x < 41 - 6\lambda \) holds for all \( x \) in the interval \( (1, 6] \), we can follow these steps:
### Step 1: Rearranging the Inequality
We start with the inequality:
\[
x^2 - 2\lambda x < 41 - 6\lambda
\]
Rearranging gives us:
\[
x^2 - 2\lambda x - (41 - 6\lambda) < 0
\]
This can be rewritten as:
\[
x^2 - 2\lambda x + (6\lambda - 41) < 0
\]
Let \( q(x) = x^2 - 2\lambda x + (6\lambda - 41) \).
### Step 2: Finding Conditions for the Quadratic to be Negative
For the quadratic \( q(x) \) to be negative for all \( x \) in the interval \( (1, 6] \), it must have no real roots and open upwards. This means:
1. The discriminant must be less than 0.
2. The value of the quadratic at the endpoints of the interval must be less than 0.
### Step 3: Discriminant Condition
The discriminant \( D \) of the quadratic \( q(x) \) is given by:
\[
D = b^2 - 4ac = (-2\lambda)^2 - 4(1)(6\lambda - 41) = 4\lambda^2 - 24\lambda + 164
\]
Setting the discriminant less than 0:
\[
4\lambda^2 - 24\lambda + 164 < 0
\]
Dividing the entire inequality by 4:
\[
\lambda^2 - 6\lambda + 41 < 0
\]
Since the discriminant of this quadratic is negative (as \( 6^2 - 4 \cdot 1 \cdot 41 < 0 \)), it does not have real roots, and thus \( \lambda^2 - 6\lambda + 41 \) is always positive.
### Step 4: Evaluating at the Endpoints
Next, we evaluate \( q(x) \) at the endpoints \( x = 1 \) and \( x = 6 \).
1. For \( x = 1 \):
\[
q(1) = 1^2 - 2\lambda(1) + (6\lambda - 41) = 1 - 2\lambda + 6\lambda - 41 = 4\lambda - 40 < 0
\]
This simplifies to:
\[
4\lambda < 40 \quad \Rightarrow \quad \lambda < 10
\]
2. For \( x = 6 \):
\[
q(6) = 6^2 - 2\lambda(6) + (6\lambda - 41) = 36 - 12\lambda + 6\lambda - 41 = -6\lambda - 5 < 0
\]
This simplifies to:
\[
-6\lambda < 5 \quad \Rightarrow \quad \lambda > -\frac{5}{6}
\]
### Step 5: Combining the Results
From the two evaluations, we have:
\[
-\frac{5}{6} < \lambda < 10
\]
### Step 6: Finding Integral Values
The integral values of \( \lambda \) that satisfy this inequality are:
\[
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
\]
This gives us a total of 10 integral values.
### Final Count
Including \( \lambda = 10 \) (since the inequality is strict), the total number of integral values of \( \lambda \) is:
\[
\text{Total integral values} = 10 - 0 + 1 = 11
\]
### Conclusion
Thus, the number of integral values of \( \lambda \) for which the inequality holds is **11**.
