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

Answer 1

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)|.

Answer 2

50