If the equation `x^(2)-4x+log_(1/2)a` = 0 does not have two distinct real roots, then maximum value of a is

Options

A. `(1)/(16)`
B. `-(1)/(4)`
C. `(1)/(4)`
D. none of these

Answer

To determine the maximum value of \( a \) such that the equation

\[ x^2 - 4x + \log_{1/2} a = 0 \]

does not have two distinct real roots, we need to analyze the conditions under which a quadratic equation has real roots. A quadratic equation of the form \( ax^2 + bx + c = 0 \) does not have two distinct real roots if its discriminant \( D \) is less than or equal to zero.

### Step 1: Identify the coefficients
In our equation, we can identify the coefficients:
- \( a = 1 \)
- \( b = -4 \)
- \( c = \log_{1/2} a \)

### Step 2: Calculate the discriminant
The discriminant \( D \) for the quadratic equation is given by:

\[ D = b^2 - 4ac \]

Substituting the values we identified:

\[ D = (-4)^2 - 4 \cdot 1 \cdot \log_{1/2} a \]

This simplifies to:

\[ D = 16 - 4 \log_{1/2} a \]

### Step 3: Set the discriminant condition
For the quadratic equation to not have two distinct real roots, we need:

\[ D \leq 0 \]

Substituting our expression for \( D \):

\[ 16 - 4 \log_{1/2} a \leq 0 \]

### Step 4: Solve the inequality
Rearranging the inequality gives:

\[ 4 \log_{1/2} a \geq 16 \]

Dividing both sides by 4:

\[ \log_{1/2} a \geq 4 \]

### Step 5: Change of base for logarithm
Using the change of base formula, we can express \( \log_{1/2} a \) in terms of the natural logarithm:

\[ \log_{1/2} a = \frac{\log a}{\log (1/2)} = \frac{\log a}{-\log 2} \]

Thus, the inequality becomes:

\[ \frac{\log a}{-\log 2} \geq 4 \]

Multiplying both sides by \( -\log 2 \) (note that this reverses the inequality):

\[ \log a \leq -4 \log 2 \]

### Step 6: Exponentiate to solve for \( a \)
Exponentiating both sides gives:

\[ a \leq 2^{-4} \]

Calculating \( 2^{-4} \):

\[ 2^{-4} = \frac{1}{16} \]

### Conclusion
Thus, the maximum value of \( a \) such that the equation does not have two distinct real roots is:

\[ \boxed{\frac{1}{16}} \]