Let `I_(1) : (log_(x)2) (log_(2x)2) (log_(2)4x)gt1`
`I_(2) : x^((log_(10)x)^(2)-3(log_(10)x)+1) gt 1000`
and solution of inequality `I_(1)` is `((1)/(a^(sqrt(a))),(1)/(b))cup(c, a^(sqrt(a)))`
and solution of inequality `I_(2)` is `(d, oo)` then answer the following
Both root of equation `dx^(2) - bx + k = 0, (k in R)` are positive then `k` can not be

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. `1`
D. `2`

Explanation

To solve the given problem, we need to analyze the inequalities \( I_1 \) and \( I_2 \) and determine the conditions under which both roots of the quadratic equation \( dx^2 - bx + k = 0 \) are positive.

### Step 1: Analyze Inequality \( I_1 \)

The first inequality is given as:

\[ I_1: (\log_x 2)(\log_{2x} 2)(\log_2 4x) > 1 \]

Let's rewrite the logarithmic expressions:

1. \( \log_x 2 = \frac{1}{\log_2 x} \)
2. \( \log_{2x} 2 = \frac{1}{\log_2 (2x)} = \frac{1}{\log_2 2 + \log_2 x} = \frac{1}{1 + \log_2 x} \)
3. \( \log_2 4x = \log_2 4 + \log_2 x = 2 + \log_2 x \)

Substituting these into the inequality:

\[ \left(\frac{1}{\log_2 x}\right) \left(\frac{1}{1 + \log_2 x}\right) (2 + \log_2 x) > 1 \]

### Step 2: Simplify the Inequality

Let \( t = \log_2 x \). The inequality becomes:

\[ \frac{(2 + t)}{t(1 + t)} > 1 \]

Rearranging gives:

\[ 2 + t > t(1 + t) \]

This simplifies to:

\[ 2 + t > t + t^2 \implies t^2 - 2 < 0 \]

### Step 3: Solve the Quadratic Inequality

The roots of the equation \( t^2 - 2 = 0 \) are \( t = -\sqrt{2} \) and \( t = \sqrt{2} \). The quadratic opens upwards, so the solution for \( t^2 - 2 < 0 \) is:

\[ -\sqrt{2} < t < \sqrt{2} \]

### Step 4: Convert Back to \( x \)

Since \( t = \log_2 x \):

\[ -\sqrt{2} < \log_2 x < \sqrt{2} \]

This translates to:

\[ 2^{-\sqrt{2}} < x < 2^{\sqrt{2}} \]

### Step 5: Analyze Inequality \( I_2 \)

The second inequality is:

\[ I_2: x^{(\log_{10} x)^2 - 3\log_{10} x + 1} > 1000 \]

Taking logarithm base 10 on both sides gives:

\[ (\log_{10} x)^2 - 3\log_{10} x + 1 > 3 \]

### Step 6: Rearranging the Inequality

Rearranging gives:

\[ (\log_{10} x)^2 - 3\log_{10} x - 2 > 0 \]

### Step 7: Solve the Quadratic Inequality

The roots of the equation \( t^2 - 3t - 2 = 0 \) can be found using the quadratic formula:

\[ t = \frac{3 \pm \sqrt{3^2 + 8}}{2} = \frac{3 \pm \sqrt{17}}{2} \]

Let \( d_1 = \frac{3 - \sqrt{17}}{2} \) and \( d_2 = \frac{3 + \sqrt{17}}{2} \). The solution for the inequality \( t^2 - 3t - 2 > 0 \) is:

\[ t < d_1 \quad \text{or} \quad t > d_2 \]

### Step 8: Determine Conditions for Positive Roots

For the quadratic \( dx^2 - bx + k = 0 \) to have both roots positive, the following conditions must hold:

1. \( d > 0 \) (leading coefficient must be positive)
2. \( k > 0 \) (constant term must be positive)
3. The discriminant \( b^2 - 4dk > 0 \)

### Step 9: Identify the Value of \( k \)

From the analysis, we find that if both roots are positive, \( k \) must be less than or equal to \( \frac{1}{1000} \). Therefore, \( k \) cannot be equal to:

\[ \text{Answer: } k = 0 \]