If `lim_(t to a) (int_(a)^(t) f(t)dt-(t-a)/2 (f(t) -f(a)))/(t-a)^(3)= 0`, then maximum degree of `f(x)` is:

Options

A. 4
B. 3
C. 2
D. 1

Answer

To solve the problem, we need to analyze the limit given in the question:

\[ \lim_{t \to a} \frac{\int_a^t f(t) \, dt - \frac{(t-a)}{2} (f(t) - f(a))}{(t-a)^3} = 0 \]

### Step 1: Identify the form of the limit
As \( t \to a \), both the numerator and denominator approach 0, which indicates that we can apply L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule
We differentiate the numerator and denominator with respect to \( t \):

1. **Numerator**: Differentiate \( \int_a^t f(t) \, dt \) with respect to \( t \):
\[ \frac{d}{dt} \left( \int_a^t f(t) \, dt \right) = f(t) \]

Differentiate \( -\frac{(t-a)}{2}(f(t) - f(a)) \):
Using the product rule:
\[ -\frac{1}{2} \left( (f(t) - f(a)) + (t-a) f'(t) \right) \]

Therefore, the derivative of the numerator becomes:
\[ f(t) - \frac{1}{2} \left( (f(t) - f(a)) + (t-a) f'(t) \right) \]

2. **Denominator**: The derivative of \( (t-a)^3 \) is:
\[ 3(t-a)^2 \]

### Step 3: Rewrite the limit
Now we can rewrite the limit using L'Hôpital's Rule:
\[ \lim_{t \to a} \frac{f(t) - \frac{1}{2} \left( (f(t) - f(a)) + (t-a) f'(t) \right)}{3(t-a)^2} \]

### Step 4: Evaluate the limit again
As \( t \to a \), we again get a \( 0/0 \) form, so we apply L'Hôpital's Rule again.

1. Differentiate the new numerator:
- The derivative of \( f(t) \) is \( f'(t) \).
- The derivative of \( -\frac{1}{2} \left( (f(t) - f(a)) + (t-a) f'(t) \right) \) involves using the product rule again.

2. Differentiate the denominator:
- The derivative of \( 3(t-a)^2 \) is \( 6(t-a) \).

### Step 5: Evaluate the limit again
This will lead to another \( 0/0 \) form, so we apply L'Hôpital's Rule once more.

### Step 6: Final evaluation
After applying L'Hôpital's Rule three times, we will eventually reach a limit involving \( f''(a) \) or higher derivatives. Since the limit is given to be zero, we conclude that:

\[ f''(a) = 0 \]

This implies that \( f'(t) \) must be a constant, and thus \( f(t) \) is a linear function.

### Conclusion
The maximum degree of \( f(x) \) is 1, which corresponds to a linear function.

### Final Answer
The maximum degree of \( f(x) \) is **1**.