If `int_(0)^(t^(2)) xf (x) dx= (2)/(5) t^(5)`, then f(4/25)=

Options

A. `-2//5`
B. `-5//2`
C. 1
D. `5//2`

Answer

To solve the problem, we need to find the value of \( f\left(\frac{4}{25}\right) \) given the equation:

\[ \int_{0}^{t^2} x f(x) \, dx = \frac{2}{5} t^5 \]

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

We start by differentiating both sides of the equation with respect to \( t \):

\[ \frac{d}{dt} \left( \int_{0}^{t^2} x f(x) \, dx \right) = \frac{d}{dt} \left( \frac{2}{5} t^5 \right) \]

### Step 2: Apply the Fundamental Theorem of Calculus

Using the Fundamental Theorem of Calculus and the chain rule, we differentiate the left-hand side:

\[ \frac{d}{dt} \left( \int_{0}^{t^2} x f(x) \, dx \right) = t^2 f(t^2) \cdot \frac{d}{dt}(t^2) - 0 = t^2 f(t^2) \cdot 2t \]

So, we have:

\[ 2t^3 f(t^2) \]

### Step 3: Differentiate the right-hand side

Now we differentiate the right-hand side:

\[ \frac{d}{dt} \left( \frac{2}{5} t^5 \right) = \frac{2}{5} \cdot 5 t^4 = 2 t^4 \]

### Step 4: Set the derivatives equal to each other

Now we equate the two results:

\[ 2t^3 f(t^2) = 2t^4 \]

### Step 5: Simplify the equation

We can simplify this by dividing both sides by \( 2t^3 \) (assuming \( t \neq 0 \)):

\[ f(t^2) = t \]

### Step 6: Substitute \( t^2 = \frac{4}{25} \)

Now, we need to find \( f\left(\frac{4}{25}\right) \). We set \( t^2 = \frac{4}{25} \), which gives us:

\[ t = \sqrt{\frac{4}{25}} = \frac{2}{5} \]

### Step 7: Find \( f\left(\frac{4}{25}\right) \)

Using the relation \( f(t^2) = t \):

\[ f\left(\frac{4}{25}\right) = \frac{2}{5} \]

### Conclusion

Thus, the value of \( f\left(\frac{4}{25}\right) \) is:

\[ \boxed{\frac{2}{5}} \]