Let f(x) be a differentiable function such that `int_(t)^(t^(2))xf(x)dx=(4)/(3)t^(3)-(4t)/(3)AA t ge0`, then f(1) is equal to

Options

A. 4
B. `(4)/(3)`
C. 3
D. `(8)/(3)`

Answer

To solve the problem, we will use the Leibniz rule for differentiating under the integral sign. The given equation is:

\[ \int_{t}^{t^2} x f(x) \, dx = \frac{4}{3} t^3 - \frac{4}{3} t \]

We need to find \( f(1) \).

### Step-by-step Solution:

1. **Differentiate both sides with respect to \( t \)**:
Using the Leibniz rule, we differentiate the left-hand side:
\[ \frac{d}{dt} \left( \int_{t}^{t^2} x f(x) \, dx \right) = t^2 f(t^2) \cdot \frac{d}{dt}(t^2) - t f(t) \cdot \frac{d}{dt}(t) \]
This simplifies to:
\[ t^2 f(t^2) \cdot 2t - t f(t) \cdot 1 = 2t^3 f(t^2) - t f(t) \]

2. **Differentiate the right-hand side**:
Now, differentiate the right-hand side:
\[ \frac{d}{dt} \left( \frac{4}{3} t^3 - \frac{4}{3} t \right) = 4t^2 - \frac{4}{3} \]

3. **Set the derivatives equal**:
Now we set the derivatives from both sides equal to each other:
\[ 2t^3 f(t^2) - t f(t) = 4t^2 - \frac{4}{3} \]

4. **Substitute \( t = 1 \)**:
To find \( f(1) \), substitute \( t = 1 \):
\[ 2(1)^3 f(1^2) - 1 f(1) = 4(1)^2 - \frac{4}{3} \]
This simplifies to:
\[ 2f(1) - f(1) = 4 - \frac{4}{3} \]
Which simplifies to:
\[ f(1) = 4 - \frac{4}{3} \]

5. **Simplify the right-hand side**:
Now, calculate \( 4 - \frac{4}{3} \):
\[ 4 = \frac{12}{3} \quad \text{(converting 4 to a fraction with a denominator of 3)} \]
Thus:
\[ f(1) = \frac{12}{3} - \frac{4}{3} = \frac{8}{3} \]

### Conclusion:
The value of \( f(1) \) is:
\[ \boxed{\frac{8}{3}} \]