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

Not too sure the point of the question, but here's the answer anyway, assuming b and c are given but unknown constants.

To find (max|f(x)|) for x∈[-10,10]:

Complete squares to get
f(x)=(x+b/2)²+c-b²/4
where (-b/2, c-b²/4) is the vertex.

If
-b/2 ∈ [-10,0] (vertex left of origin) then
max |f(x)| = max(|f(-b/2)|,|f(10)|)

if
-b/2 ∈ [0,10] (vertex right of origin) then
max |f(x)| = max(|f(-b/2)|,|f(-10)|)

Otherwise
max|f(x)| = max(|f(-10)|,|f(10)|)

Not sure what is meant by:
minimum of max|f(x)|.

50

To find the minimum value of the maximum absolute value of a function within a given interval, we need to first find the critical points of the function within that interval.

Given the function f(x) = x^2 + bx + c, where b and c are real numbers, we want to find the maximum |f(x)| for x ∈ [-10, 10].

Step 1: Find the critical points of f(x)
To find the critical points, we need to find the values of x for which the derivative of f(x) is equal to zero. The critical points will occur at the turning points of the function.

Taking the derivative of f(x) with respect to x:
f'(x) = 2x + b

Setting f'(x) equal to zero and solving for x:
2x + b = 0
x = -b/2

The critical point of f(x) is x = -b/2.

Step 2: Calculate the maximum |f(x)|
To find the maximum |f(x)| within the interval [-10, 10], we evaluate the function at the critical point and at the endpoints of the interval.

Evaluate f(x) at x = -10, x = 10, and x = -b/2.
f(-10) = (-10)^2 + b(-10) + c
f(10) = (10)^2 + b(10) + c
f(-b/2) = (-b/2)^2 + b(-b/2) + c

Step 3: Find the maximum absolute value
Compare the absolute values of the function evaluations at the three points calculated in step 2 (|f(-10)|, |f(10)|, and |f(-b/2)|), and determine which one is the maximum.

max{|f(x)|} = max{|f(-10)|, |f(10)|, |f(-b/2)|}

Step 4: Calculate the minimum value
Evaluate the maximum absolute value found in step 3 for different values of b and c in order to find the minimum of all the maximum absolute values.

By plugging in different values of b and c, we can find the minimum value of the maximum absolute value of the function.

It's important to note that the minimum value of the maximum |f(x)| within the given interval will vary depending on the values of b and c.