f(x)=x ^2 +bx+c , b and c real .Find the minimum of (max|f(x)|)for x∈[−10,10]?

To find the minimum value of the maximum absolute value of the function f(x) = x^2 + bx + c for x ∈ [-10, 10], we can follow these steps:

1. Calculate the maximum and minimum values of the function f(x) in the given range [-10, 10]. To find the extreme values of a quadratic function, consider its vertex.

The vertex of a quadratic function in the form f(x) = ax^2 + bx + c is given by the formula:
x = -b / (2a)
y = f(x)

In this case, a = 1, so the x-coordinate of the vertex is x = -b / 2.

2. Plug in the x-coordinate of the vertex into the equation f(x) to get the corresponding y-coordinate.

3. Calculate the absolute values of the y-coordinate at the vertex and the maximum and minimum values obtained in step 1.

4. Compare the absolute values obtained in step 3 and find the minimum value among them.

Let's calculate the minimum value of the maximum absolute value of f(x):

Step 1:
The x-coordinate of the vertex, x = -b / 2

Step 2:
Plug in x = -b / 2 into the function f(x) to get the y-coordinate:
y = (-b / 2)^2 + b(-b / 2) + c
y = b^2 / 4 - b^2 / 2 + c
y = c - b^2 / 4

Step 3:
Calculate the absolute values of the y-coordinate at the vertex and the maximum and minimum values in the given range [-10, 10].
- At the vertex: |c - b^2 / 4|
- At x = -10: |(-10)^2 + b(-10) + c| = |100 - 10b + c|
- At x = 10: |10^2 + b(10) + c| = |100 + 10b + c|

Step 4:
Compare the absolute values and find the minimum value among them:
minimum = min(|c - b^2 / 4|, |100 - 10b + c|, |100 + 10b + c|)

By following these steps, you can find the minimum value of the maximum absolute value of the function f(x) = x^2 + bx + c for x ∈ [-10, 10].