Let `f(x)=x^(2) +ax + b` and the only solution of the equation `f(x)= min(f(x)) is x =0 and f(x) =0 "has root " alpha and beta, "then" int _(alpha )^(beta)x^(3) dx` is equal to

Options

A. `(1)/(4) ( beta ^(4) + alpha ^(4))`
B. `(1)/(4)( a^(2)- b^(2))`
C. 0
D. None of these

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understanding the Function
Given the function \( f(x) = x^2 + ax + b \), we know that it has a minimum at \( x = 0 \). This implies that the derivative of the function at this point must be zero.

### Step 2: Finding the Derivative
We differentiate \( f(x) \):
\[ f'(x) = 2x + a \]
Setting the derivative equal to zero at \( x = 0 \):
\[ f'(0) = 2(0) + a = 0 \implies a = 0 \]

### Step 3: Simplifying the Function
Substituting \( a = 0 \) back into the function:
\[ f(x) = x^2 + b \]

### Step 4: Finding the Roots
The function \( f(x) = 0 \) has roots \( \alpha \) and \( \beta \). Setting the function equal to zero:
\[ x^2 + b = 0 \implies x^2 = -b \]
This means the roots are:
\[ \alpha = i\sqrt{-b}, \quad \beta = -i\sqrt{-b} \]
Thus, the roots are purely imaginary unless \( b \) is negative.

### Step 5: Finding the Sum and Product of Roots
From the quadratic formula, we know:
- The sum of the roots \( \alpha + \beta = 0 \)
- The product of the roots \( \alpha \beta = b \)

### Step 6: Setting Up the Integral
We need to evaluate the integral:
\[ \int_{\alpha}^{\beta} x^3 \, dx \]

### Step 7: Evaluating the Integral
The integral of \( x^3 \) is:
\[ \int x^3 \, dx = \frac{x^4}{4} \]
Now we evaluate it from \( \alpha \) to \( \beta \):
\[ \int_{\alpha}^{\beta} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{\alpha}^{\beta} = \frac{\beta^4}{4} - \frac{\alpha^4}{4} \]
Factoring out \( \frac{1}{4} \):
\[ = \frac{1}{4} (\beta^4 - \alpha^4) \]

### Step 8: Using the Difference of Squares
Using the identity \( a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) \):
\[ \beta^4 - \alpha^4 = (\beta^2 - \alpha^2)(\beta^2 + \alpha^2) \]
And since \( \beta = -\alpha \):
\[ \beta^2 - \alpha^2 = 0 \implies \beta^4 - \alpha^4 = 0 \]

### Step 9: Conclusion
Thus, we have:
\[ \int_{\alpha}^{\beta} x^3 \, dx = \frac{1}{4} \cdot 0 = 0 \]

### Final Answer
The value of the integral is:
\[ \boxed{0} \]