Let `I=int_(a)^(b)(x^(4)-2x^(2))dx`. If I is minimum then the ordered pair (a, b) is:

Options

A. `(0, sqrt(2))`
B. `(-sqrt(2),sqrt(2))`
C. `(sqrt(2), -sqrt(2))`
D. `(-sqrt(2), 0)`

Answer

To solve the problem, we need to find the ordered pair \((a, b)\) such that the integral \(I = \int_{a}^{b} (x^4 - 2x^2) \, dx\) is minimized.

### Step-by-Step Solution:

1. **Identify the Function**:
We start with the function \(f(x) = x^4 - 2x^2\).

2. **Find Critical Points**:
To find the minimum value of the integral, we first need to find the critical points of the function \(f(x)\). We do this by taking the derivative and setting it to zero.
\[ f'(x) = \frac{d}{dx}(x^4 - 2x^2) = 4x^3 - 4x \]
Setting the derivative equal to zero:
\[ 4x^3 - 4x = 0 \implies 4x(x^2 - 1) = 0 \]
This gives us:
\[ x = 0, \quad x = 1, \quad x = -1 \]

3. **Evaluate the Function at Critical Points**:
We evaluate \(f(x)\) at the critical points:
\[ f(0) = 0^4 - 2(0^2) = 0 \]
\[ f(1) = 1^4 - 2(1^2) = 1 - 2 = -1 \]
\[ f(-1) = (-1)^4 - 2(-1)^2 = 1 - 2 = -1 \]

4. **Determine the Minimum Value**:
The minimum value of \(f(x)\) occurs at \(x = 1\) and \(x = -1\) where \(f(1) = f(-1) = -1\).

5. **Set the Limits of Integration**:
To minimize the integral \(I\), we should integrate from the minimum points found:
\[ a = -1, \quad b = 1 \]

6. **Conclusion**:
The ordered pair \((a, b)\) that minimizes the integral \(I\) is:
\[ (a, b) = (-1, 1) \]

### Final Answer:
The ordered pair \((a, b)\) is \((-1, 1)\).