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 start with the given integral equation:

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

for \( t \geq 0 \). We need to find \( f(1) \).

### Step 1: Differentiate both sides with respect to \( t \)

Using the Fundamental Theorem of Calculus and the Leibniz rule for differentiation under the integral sign, we differentiate the left-hand side:

\[ \frac{d}{dt} \left( \int_{t}^{t^2} x f(x) \, dx \right) = t^2 f(t^2) \cdot \frac{d}{dt}(t^2) - t f(t) \cdot \frac{d}{dt}(t) = 2t^2 f(t^2) - t f(t) \]

Now differentiate the right-hand side:

\[ \frac{d}{dt} \left( \frac{4}{3} t^3 - \frac{4}{3} t \right) = 4t^2 - \frac{4}{3} \]

### Step 2: Set the derivatives equal to each other

We equate the derivatives from both sides:

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

### Step 3: Rearrange the equation

Rearranging gives:

\[ 2t^2 f(t^2) = t f(t) + 4t^2 - \frac{4}{3} \]

### Step 4: Solve for \( f(t) \)

Now, we will isolate \( f(t) \):

\[ f(t) = \frac{2t^2 f(t^2) - 4t^2 + \frac{4}{3}}{t} \]

### Step 5: Evaluate at \( t = 1 \)

To find \( f(1) \), we substitute \( t = 1 \):

\[ f(1) = 2 \cdot 1^2 f(1^2) - 4 \cdot 1^2 + \frac{4}{3} \]

This simplifies to:

\[ f(1) = 2f(1) - 4 + \frac{4}{3} \]

### Step 6: Solve for \( f(1) \)

Rearranging gives:

\[ f(1) - 2f(1) = -4 + \frac{4}{3} \]

This simplifies to:

\[ -f(1) = -4 + \frac{4}{3} \]

Converting \(-4\) to a fraction gives:

\[ -4 = -\frac{12}{3} \]

Thus:

\[ -f(1) = -\frac{12}{3} + \frac{4}{3} = -\frac{8}{3} \]

So:

\[ f(1) = \frac{8}{3} \]

### Final Answer

Thus, the value of \( f(1) \) is:

\[ \boxed{\frac{8}{3}} \]