If `C_0/1+C_1/2+C_2/3=0` , where `C_0 C_1, C_2` are all real, the equation `C_2x^2+C_1x+C_0=0` has

Options

A. at least one root in (0,1)
B. one root in (1,2) and the other In(3,4)
C. one root in (-1,1) and the other in (-5,2)
D. both roots imaginary

Answer

To solve the problem, we need to analyze the equation \( C_2 x^2 + C_1 x + C_0 = 0 \) under the condition that \( \frac{C_0}{1} + \frac{C_1}{2} + \frac{C_2}{3} = 0 \).

### Step 1: Understanding the condition
Given the equation:
\[ \frac{C_0}{1} + \frac{C_1}{2} + \frac{C_2}{3} = 0 \]
This can be rewritten as:
\[ C_0 + \frac{C_1}{2} + \frac{C_2}{3} = 0 \]
This implies that the weighted sum of the coefficients \( C_0, C_1, \) and \( C_2 \) equals zero.

### Step 2: Setting up the function
Let:
\[ f(x) = C_2 x^2 + C_1 x + C_0 \]
We need to determine the roots of \( f(x) = 0 \).

### Step 3: Applying the Intermediate Value Theorem
To apply the Intermediate Value Theorem, we need to evaluate the integral of \( f(x) \) over specific intervals. If the integral of \( f(x) \) over an interval is zero, then \( f(x) \) must have at least one root in that interval.

### Step 4: Evaluating the integral from 0 to 1
We calculate:
\[ \int_0^1 f(x) \, dx = \int_0^1 (C_2 x^2 + C_1 x + C_0) \, dx \]
Calculating the integral:
\[ = \left[ \frac{C_2 x^3}{3} + \frac{C_1 x^2}{2} + C_0 x \right]_0^1 \]
Substituting the limits:
\[ = \left( \frac{C_2}{3} + \frac{C_1}{2} + C_0 \right) - 0 \]
Using the condition \( C_0 + \frac{C_1}{2} + \frac{C_2}{3} = 0 \):
\[ = 0 \]
Thus, \( f(x) \) has at least one root in the interval \( [0, 1] \).

### Step 5: Evaluating the integral from 1 to 2
Now, we evaluate:
\[ \int_1^2 f(x) \, dx = \int_1^2 (C_2 x^2 + C_1 x + C_0) \, dx \]
Calculating the integral:
\[ = \left[ \frac{C_2 x^3}{3} + \frac{C_1 x^2}{2} + C_0 x \right]_1^2 \]
Substituting the limits:
\[ = \left( \frac{C_2 (2^3)}{3} + \frac{C_1 (2^2)}{2} + C_0 (2) \right) - \left( \frac{C_2 (1^3)}{3} + \frac{C_1 (1^2)}{2} + C_0 (1) \right) \]
This simplifies to:
\[ = \left( \frac{8C_2}{3} + 2C_1 + 2C_0 \right) - \left( \frac{C_2}{3} + \frac{C_1}{2} + C_0 \right) \]
Combining the terms:
\[ = \left( \frac{8C_2}{3} - \frac{C_2}{3} \right) + \left( 2C_1 - \frac{C_1}{2} \right) + (2C_0 - C_0) \]
\[ = \frac{7C_2}{3} + \frac{4C_1}{2} + C_0 \]
This does not equal zero, indicating that \( f(x) \) does not have a root in the interval \( [1, 2] \).

### Step 6: Conclusion
From the analysis, we find that the equation \( C_2 x^2 + C_1 x + C_0 = 0 \) has at least one root in the interval \( [0, 1] \) but does not have a root in the interval \( [1, 2] \). Therefore, the final conclusion is that the equation has:

**At least one root in the interval [0, 1] and no roots in the interval [1, 2].**