Let `f:[1, oo[ to [2, oo[` be differentiable function such that f(1)=2. If `6int_(1)^(x) f(t)dt=3x f(x)-x^(3)-5` for all `x ge 1`, then the value of f(2) is

Options

A. 6
B. 4
C. `(1)/(4)`
D. `(1)/(6)`

Answer

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

\[ 6 \int_{1}^{x} f(t) \, dt = 3x f(x) - x^3 - 5 \]

for all \( x \geq 1 \).

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

Using Leibniz's rule for differentiation under the integral sign, we differentiate the left-hand side:

\[ \frac{d}{dx} \left( 6 \int_{1}^{x} f(t) \, dt \right) = 6 f(x) \]

Now, differentiate the right-hand side:

\[ \frac{d}{dx} \left( 3x f(x) - x^3 - 5 \right) = 3 f(x) + 3x f'(x) - 3x^2 \]

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

Equating the derivatives from both sides gives us:

\[ 6 f(x) = 3 f(x) + 3x f'(x) - 3x^2 \]

### Step 3: Rearrange the equation

Rearranging the equation, we have:

\[ 6 f(x) - 3 f(x) + 3x^2 = 3x f'(x) \]

This simplifies to:

\[ 3 f(x) + 3x^2 = 3x f'(x) \]

Dividing through by 3:

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

### Step 4: Rewrite the equation

We can rewrite this as:

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

### Step 5: Identify the form of the differential equation

This is a linear first-order differential equation of the form:

\[ f'(x) - \frac{f(x)}{x} = x \]

### Step 6: Find the integrating factor

The integrating factor \( \mu(x) \) is given by:

\[ \mu(x) = e^{\int -\frac{1}{x} \, dx} = e^{-\ln |x|} = \frac{1}{x} \]

### Step 7: Multiply through by the integrating factor

Multiplying the entire differential equation by \( \frac{1}{x} \):

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

### Step 8: Integrate both sides

Now we integrate both sides:

\[ \int \left( \frac{f'(x)}{x} - \frac{f(x)}{x^2} \right) dx = \int 1 \, dx \]

The left-hand side can be rewritten as:

\[ \frac{f(x)}{x} = x + C \]

where \( C \) is the constant of integration.

### Step 9: Solve for \( f(x) \)

Multiplying through by \( x \):

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

### Step 10: Use the initial condition

We know that \( f(1) = 2 \):

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

This gives:

\[ 1 + C = 2 \implies C = 1 \]

### Step 11: Write the final form of \( f(x) \)

Thus, we have:

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

### Step 12: Find \( f(2) \)

Now, we find \( f(2) \):

\[ f(2) = 2^2 + 2 = 4 + 2 = 6 \]

### Final Answer

The value of \( f(2) \) is:

\[ \boxed{6} \]

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