If `f(x)` is differentiable function in the interval `(0,oo)` such that f(1) = 1 and `lim_(trarrx) (t^(2)f(x)-x^(2)f(t))/(t-x)=1` for each `x gt 0`, then `f((3)/(2))` is equal to

Options

A. `(13)/(6)`
B. `(23)/(18)`
C. `(25)/(9)`
D. `(31)/(18)`

Explanation

To solve the problem, we need to analyze the given limit and the properties of the function \( f(x) \). Let's break it down step by step.

### Step 1: Analyze the Limit
We are given the limit:
\[ \lim_{t \to x} \frac{t^2 f(x) - x^2 f(t)}{t - x} = 1 \]
This limit can be interpreted as the derivative of a function. Specifically, we can rewrite it as:
\[ \lim_{t \to x} \frac{t^2 f(x) - x^2 f(t)}{t - x} \]
This is a \( \frac{0}{0} \) form when we substitute \( t = x \), so we can apply L'Hôpital's rule.

### Step 2: Apply L'Hôpital's Rule
Using L'Hôpital's rule, we differentiate the numerator and the denominator with respect to \( t \):
- The numerator: \( \frac{d}{dt}(t^2 f(x) - x^2 f(t)) = 2t f(x) - x^2 f'(t) \)
- The denominator: \( \frac{d}{dt}(t - x) = 1 \)

Thus, we have:
\[ \lim_{t \to x} (2t f(x) - x^2 f'(t)) = 1 \]

### Step 3: Substitute \( t = x \)
Now, substituting \( t = x \):
\[ 2x f(x) - x^2 f'(x) = 1 \]

### Step 4: Rearranging the Equation
Rearranging gives us:
\[ x^2 f'(x) - 2x f(x) + 1 = 0 \]
This is a first-order linear differential equation.

### Step 5: Rewrite in Standard Form
We can rewrite this as:
\[ f'(x) - \frac{2}{x} f(x) = \frac{1}{x^2} \]

### Step 6: Find the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int -\frac{2}{x} \, dx} = e^{-2 \ln x} = \frac{1}{x^2} \]

### Step 7: Multiply the Equation by the Integrating Factor
Multiplying the entire differential equation by \( \frac{1}{x^2} \):
\[ \frac{1}{x^2} f'(x) - \frac{2}{x^3} f(x) = \frac{1}{x^4} \]

### Step 8: Integrate Both Sides
The left-hand side can be expressed as the derivative of a product:
\[ \frac{d}{dx}\left(\frac{f(x)}{x^2}\right) = \frac{1}{x^4} \]
Integrating both sides:
\[ \frac{f(x)}{x^2} = -\frac{1}{3x^3} + C \]
where \( C \) is the constant of integration.

### Step 9: Solve for \( f(x) \)
Multiplying through by \( x^2 \):
\[ f(x) = -\frac{1}{3x} + Cx^2 \]

### Step 10: Use Initial Condition
We know \( f(1) = 1 \):
\[ 1 = -\frac{1}{3 \cdot 1} + C \cdot 1^2 \implies 1 = -\frac{1}{3} + C \implies C = 1 + \frac{1}{3} = \frac{4}{3} \]

### Step 11: Final Form of \( f(x) \)
Thus, the function is:
\[ f(x) = -\frac{1}{3x} + \frac{4}{3}x^2 \]

### Step 12: Find \( f\left(\frac{3}{2}\right) \)
Now, we need to calculate \( f\left(\frac{3}{2}\right) \):
\[ f\left(\frac{3}{2}\right) = -\frac{1}{3 \cdot \frac{3}{2}} + \frac{4}{3} \left(\frac{3}{2}\right)^2 \]
Calculating each term:
\[ -\frac{1}{3 \cdot \frac{3}{2}} = -\frac{2}{9} \]
\[ \frac{4}{3} \left(\frac{3}{2}\right)^2 = \frac{4}{3} \cdot \frac{9}{4} = 3 \]
Thus,
\[ f\left(\frac{3}{2}\right) = -\frac{2}{9} + 3 = 3 - \frac{2}{9} = \frac{27}{9} - \frac{2}{9} = \frac{25}{9} \]

### Final Answer
\[ f\left(\frac{3}{2}\right) = \frac{25}{9} \]