If `c gt 0 and 4a + c lt 2b`. Then `ax^(2) - bx + c = 0` has a root in which one of the following intervals ?

Options

A. (0,2)
B. (2,3)
C. (3,4)
D. `(-2,0)`

Answer

To solve the problem step by step, we will analyze the quadratic equation \( ax^2 - bx + c = 0 \) given the conditions \( c > 0 \) and \( 4a + c < 2b \).

### Step 1: Define the function
Let \( f(x) = ax^2 - bx + c \).

### Step 2: Evaluate \( f(0) \)
Substituting \( x = 0 \) into the function:
\[ f(0) = a(0)^2 - b(0) + c = c \]
Since \( c > 0 \), we have:
\[ f(0) > 0 \]

### Step 3: Evaluate \( f(2) \)
Now, substitute \( x = 2 \):
\[ f(2) = a(2)^2 - b(2) + c = 4a - 2b + c \]

### Step 4: Analyze the inequality
From the problem, we know:
\[ 4a + c < 2b \]
Rearranging this gives:
\[ 4a + c - 2b < 0 \]
Thus:
\[ f(2) = 4a - 2b + c < 0 \]

### Step 5: Conclusion about the roots
Since \( f(0) > 0 \) and \( f(2) < 0 \), by the Intermediate Value Theorem, there must be at least one root of the function \( f(x) = 0 \) in the interval \( (0, 2) \).

### Final Answer
The quadratic equation \( ax^2 - bx + c = 0 \) has a root in the interval \( (0, 2) \).

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