`If int_(0)^(1) f(t)dt=x^2+int_(0)^(1) t^2f(t)dt`, then f'(1/2)is

Options

A. `6/25`
B. `24/25`
C. `18/25`
D. `4/5`

Answer

To solve the problem, we start with the given equation:

\[ \int_{0}^{x} f(t) \, dt = x^2 + \int_{x}^{1} t^2 f(t) \, dt \]

We need to find \( f'(1/2) \).

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

Using the Fundamental Theorem of Calculus and Leibniz's rule for differentiating under the integral sign, we differentiate both sides:

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

For the right-hand side, we differentiate:

\[ \frac{d}{dx} \left( x^2 + \int_{x}^{1} t^2 f(t) \, dt \right) = 2x + \frac{d}{dx} \left( \int_{x}^{1} t^2 f(t) \, dt \right) \]

Using Leibniz's rule:

\[ \frac{d}{dx} \left( \int_{x}^{1} t^2 f(t) \, dt \right) = -x^2 f(x) \]

So, we have:

\[ f(x) = 2x - x^2 f(x) \]

### Step 2: Rearranging the equation

Rearranging the equation gives:

\[ f(x) + x^2 f(x) = 2x \]

Factoring out \( f(x) \):

\[ f(x)(1 + x^2) = 2x \]

### Step 3: Solve for \( f(x) \)

Now, we can solve for \( f(x) \):

\[ f(x) = \frac{2x}{1 + x^2} \]

### Step 4: Differentiate \( f(x) \)

Next, we need to find \( f'(x) \):

Using the quotient rule:

\[ f'(x) = \frac{(1 + x^2)(2) - 2x(2x)}{(1 + x^2)^2} \]

This simplifies to:

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

### Step 5: Evaluate \( f'(1/2) \)

Now, we substitute \( x = \frac{1}{2} \):

\[ f'\left(\frac{1}{2}\right) = \frac{2 - 2\left(\frac{1}{2}\right)^2}{\left(1 + \left(\frac{1}{2}\right)^2\right)^2} \]

Calculating this step-by-step:

1. Calculate \( \left(\frac{1}{2}\right)^2 = \frac{1}{4} \)
2. Substitute into the numerator: \( 2 - 2 \cdot \frac{1}{4} = 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \)
3. Calculate the denominator: \( 1 + \frac{1}{4} = \frac{5}{4} \) and then square it: \( \left(\frac{5}{4}\right)^2 = \frac{25}{16} \)

Putting it all together:

\[ f'\left(\frac{1}{2}\right) = \frac{\frac{3}{2}}{\frac{25}{16}} = \frac{3}{2} \cdot \frac{16}{25} = \frac{48}{50} = \frac{24}{25} \]

### Final Answer

Thus, \( f'\left(\frac{1}{2}\right) = \frac{24}{25} \).