Assertion (A): The maximum value of `log_(1//3)(x^(2)-4x+5)` is zero
Reason (R): `log_(a)x le 0` for `x ge 1` and `0 lt a lt 1`

Options

A. 1) Both (A) (R) are true and (R) is the correct explanation of (A)
B. 2) Both (A) and (R) are false
C. 3) (A) is true but (R) is false
D. 4) Both (A) and (R) are true but (R) is not correct explanation of (A)

Answer

To solve the problem, we need to analyze the assertion and the reason provided.

### Step 1: Understanding the Assertion
The assertion states that the maximum value of \( \log_{1/3}(x^2 - 4x + 5) \) is 0.

### Step 2: Finding the Function
We define the function:
\[ f(x) = \log_{1/3}(x^2 - 4x + 5) \]
To find the maximum value of this function, we need to analyze the expression inside the logarithm.

### Step 3: Simplifying the Quadratic Expression
The quadratic expression \( x^2 - 4x + 5 \) can be rewritten by completing the square:
\[ x^2 - 4x + 4 + 1 = (x - 2)^2 + 1 \]
This shows that \( x^2 - 4x + 5 \) is always greater than or equal to 1 for all real \( x \) since \( (x - 2)^2 \geq 0 \).

### Step 4: Finding the Maximum Value of the Logarithm
Since \( x^2 - 4x + 5 \geq 1 \), we can evaluate the logarithm:
\[ f(x) = \log_{1/3}((x - 2)^2 + 1) \]
The logarithm \( \log_{1/3}(y) \) is less than or equal to 0 when \( y \leq 1 \). Since \( (x - 2)^2 + 1 \geq 1 \), we can conclude that:
\[ f(x) \leq 0 \]
The maximum value occurs when \( (x - 2)^2 + 1 = 1 \), which happens when \( x = 2 \).

### Step 5: Evaluating the Function at \( x = 2 \)
Substituting \( x = 2 \):
\[ f(2) = \log_{1/3}(1) = 0 \]
Thus, the maximum value of \( f(x) \) is indeed 0.

### Step 6: Understanding the Reason
The reason states that \( \log_a(x) \leq 0 \) for \( x \geq 1 \) and \( 0 < a < 1 \). This is true because the logarithm function with a base less than 1 is decreasing. Since \( x^2 - 4x + 5 \geq 1 \), it follows that:
\[ \log_{1/3}(x^2 - 4x + 5) \leq 0 \]

### Conclusion
Both the assertion and the reason are correct. The assertion is true because the maximum value of \( \log_{1/3}(x^2 - 4x + 5) \) is 0, and the reason correctly explains why this is the case.

### Final Answer
Both A and R are true, and R is the correct explanation of A.

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