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The problem asks us to find the sum of all values of the parameter \( m \) such that the minimum value of the function \( y = (x^3 - 3x + m)^2 \) on the interval \([-1, 1]\) is equal to 1.

Let's break this down into steps to solve the problem:

### Step 1: Analyze the function
The given function is:
\[
y = (x^3 - 3x + m)^2
\]
The minimum value of \( y \) occurs when the expression inside the square, \( x^3 - 3x + m \), is at its minimum.

### Step 2: Find the critical points
To find the critical points of \( x^3 - 3x \), we take the derivative of \( f(x) = x^3 - 3x \):
\[
f'(x) = 3x^2 - 3
\]
Setting \( f'(x) = 0 \) to find critical points:
\[
3x^2 - 3 = 0 \implies x^2 = 1 \implies x = \pm 1
\]

### Step 3: Evaluate \( f(x) \) on the interval \([-1, 1]\)
Now, we evaluate \( f(x) = x^3 - 3x \) at the critical points and endpoints:

- At \( x = 1 \): \( f(1) = 1^3 - 3(1) = 1 - 3 = -2 \)
- At \( x = -1 \): \( f(-1) = (-1)^3 - 3(-1) = -1 + 3 = 2 \)

The values of \( f(x) \) at the critical points and endpoints are:
- \( f(1) = -2 \)
- \( f(-1) = 2 \)

### Step 4: Set the minimum value of \( y \) to 1
Since we need the minimum value of \( y \) to be 1, this means the expression inside the square, \( x^3 - 3x + m \), must have its minimum equal to \( \pm 1 \).

The minimum value of \( f(x) = x^3 - 3x \) occurs at \( x = 1 \), where \( f(1) = -2 \). To make this minimum equal to 1, we set:
\[
-2 + m = 0 \implies m = 2
\]

Thus, the value of \( m \) is 2.

### Conclusion
The sum of all values of \( m \) is \( 2 \).

Would you like further details or clarifications on any of these steps?

Here are 5 follow-up questions:
1. How do you find critical points of a cubic function?
2. What does the derivative tell us about the function’s behavior?
3. Why did we evaluate the function at the critical points and endpoints?
4. What happens if the minimum value is set to a different number instead of 1?
5. How can you generalize this process for other similar problems?

**Tip**: Always check for critical points and endpoints when finding the minimum or maximum of a function over a closed interval.

Tính tổng tất cả các giá trị của tham số m sao cho giá trị nhỏ nhất của hàm số y = (x^3 - 3x + m)^2 trên đoạn [-1, 1] bằng 1.

Discover how to find the sum of all values of parameter m such that the minimum value of y = (x^3 - 3x + m)^2 on the interval [-1, 1] equals 1. This problem involves calculus concepts like derivatives and critical points, and is suitable for high school students.

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