If `f:R in R` is continuous and differentiable function such that
`int_(-1)^(x) f(t)dt+f'''(3) int_(x)^(0) dt=int_(1)^(x) t^(3)dt-f'(1)int_(0)^(x) t^(2)dt+f'(2) int_(x)^(3) r dt`, then the value of f'(4), is

Options

A. `48-8f'(1)+f'(2)`
B. `48-8f'(1)-f''(2)`
C. `48+8f'(1)-f'(2)`
D. none of these

Explanation

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

\[ \int_{-1}^{x} f(t) dt + f'''(3) \int_{x}^{0} dt = \int_{1}^{x} t^3 dt - f'(1) \int_{0}^{x} t^2 dt + f'(2) \int_{x}^{3} t dt \]

### Step 1: Simplify the Integrals
First, we simplify the integrals on both sides of the equation.

The integral \(\int_{x}^{0} dt\) can be rewritten as \(-\int_{0}^{x} dt = -x\).

So, the equation becomes:

\[ \int_{-1}^{x} f(t) dt - f'''(3)x = \int_{1}^{x} t^3 dt - f'(1) \int_{0}^{x} t^2 dt + f'(2) \int_{x}^{3} t dt \]

### Step 2: Evaluate the Right Side Integrals
Now, we evaluate the right-hand side integrals:

1. \(\int_{1}^{x} t^3 dt = \left[\frac{t^4}{4}\right]_{1}^{x} = \frac{x^4}{4} - \frac{1}{4}\)
2. \(\int_{0}^{x} t^2 dt = \left[\frac{t^3}{3}\right]_{0}^{x} = \frac{x^3}{3}\)
3. \(\int_{x}^{3} t dt = \left[\frac{t^2}{2}\right]_{x}^{3} = \frac{9}{2} - \frac{x^2}{2}\)

Substituting these back into the equation gives:

\[ \int_{-1}^{x} f(t) dt - f'''(3)x = \left(\frac{x^4}{4} - \frac{1}{4}\right) - f'(1)\left(\frac{x^3}{3}\right) + f'(2)\left(\frac{9}{2} - \frac{x^2}{2}\right) \]

### Step 3: Rearranging the Equation
Rearranging the equation, we have:

\[ \int_{-1}^{x} f(t) dt - f'''(3)x = \frac{x^4}{4} - \frac{1}{4} - \frac{f'(1)x^3}{3} + \frac{9f'(2)}{2} - \frac{f'(2)x^2}{2} \]

### Step 4: Differentiate Both Sides
Next, we differentiate both sides with respect to \(x\):

Using the Fundamental Theorem of Calculus and Leibniz's rule, we differentiate:

\[ f(x) - f'''(3) = x^3 - f'(1)x^2 + \frac{9f'(2)}{2} - f'(2)x \]

### Step 5: Further Differentiate
Now, we differentiate again:

\[ f'(x) = 3x^2 - 2f'(1)x - f'(2) \]

### Step 6: Substitute \(x = 4\)
Now we substitute \(x = 4\) to find \(f'(4)\):

\[ f'(4) = 3(4^2) - 2f'(1)(4) - f'(2) \]

Calculating \(f'(4)\):

\[ f'(4) = 3(16) - 8f'(1) - f'(2) = 48 - 8f'(1) - f'(2) \]

### Final Answer
Thus, the value of \(f'(4)\) is:

\[ f'(4) = 48 - 8f'(1) - f'(2) \]