The derivative of the function $$ f $$ is defined by $$ f^{\prime}(x)=x^{2} \sin \left(x^{2}-x\right) $$. If $$ f(2)=3 $$, what is the absolute minimum value of the function $$ f $$ on the closed interval $$ [-2,2] $$ ? You may use a calculator and round your answer to the nearest thousandth.

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It seems that I cannot perform numerical integration directly. Instead, I will outline the steps to evaluate $$ f(-2) $$ and $$ f(0) $$ using numerical methods, and then we can summarize the results.

### Step 1: Evaluate $$ f(-2) $$

To find $$ f(-2) $$, we can use numerical integration of $$ f'(x) $$ from $$ -2 $$ to $$ 2 $$:

$$
f(-2) = f(2) - \int_{-2}^{2} f'(x) \, dx
$$

### Step 2: Evaluate $$ f(0) $$

To find $$ f(0) $$, we can use numerical integration of $$ f'(x) $$ from $$ 0 $$ to $$ 2 $$:

$$
f(0) = f(2) - \int_{0}^{2} f'(x) \, dx
$$

### Step 3: Numerical Integration

You can use a calculator or software to compute the following integrals:

1. $$\int_{-2}^{2} x^{2} \sin(x^{2}-x) \, dx$$
2. $$\int_{0}^{2} x^{2} \sin(x^{2}-x) \, dx$$

### Step 4: Summary of Values

Once you have the values of the integrals, you can compute:

- $$ f(-2) $$
- $$ f(0) $$
- $$ f(2) = 3 $$

### Step 5: Find the Minimum Value

Finally, compare $$ f(-2) $$, $$ f(0) $$, and $$ f(2) $$ to find the absolute minimum value of $$ f $$ on the interval $$ [-2, 2] $$.

If you can perform the numerical integration, please provide the results, and I can help you find the minimum value. Alternatively, if you need assistance with the numerical integration process, let me know!
To find the absolute minimum value of $$ f $$ on $$[-2, 2]$$, evaluate $$ f $$ at critical points and endpoints, then compare the values.

The derivative of the function is defined by . If
, what is the absolute minimum value of the function on the closed
interval ? You may use a calculator and round your answer to the nearest
thousandth.

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The derivative of the function is defined by . If , what is the absolute minimum value of the function on the closed interval ? You may use a calculator and round your answer to the nearest thousandth.