f(x)=x ^2 +bx+c , b and c real .Find the minimum of (max|f(x)|)for x∈[−10,10]?
f(x) = x^2 - 100
has maximum value of 0 in [10,10]
For any b,c note equal to (0,-100), the max of f(x) will be nonzero.
So, min(max(|f(x)|)) = 0
50
To find the minimum of the maximum value of |f(x)| for x∈[−10,10], we need to analyze the behavior of the function f(x) = x^2 + bx + c.
Step 1: Finding the maximum value of |f(x)|
Since the function f(x) is a quadratic, we can start by examining the quadratic equation f(x) = x^2 + bx + c.
To find the maximum value of |f(x)|, we need to find the vertex of the parabolic curve represented by the equation f(x).
The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a is the coefficient of the x^2 term (which is 1 in this case).
So, x = -b / (2 * 1) = -b / 2.
Now, plugging this value of x into the equation f(x) = x^2 + bx + c, we get f(-b/2) = (-b/2)^2 + b(-b/2) + c = b^2/4 - b^2/2 + c = (c - b^2/4).
Step 2: Finding the minimum of (max|f(x)|)
To find the minimum of (max|f(x)|), we need to find the minimum value of |f(x)| for all x ∈ [-10,10].
Since we found that the maximum value of |f(x)| is (c - b^2/4), the minimum of this maximum value occurs when the expression (c - b^2/4) is minimized.
As c and b are real constants, this minimum value is simply the value of (c - b^2/4).
Therefore, to find the minimum of (max|f(x)|) for x∈[−10,10], we need to minimize the expression (c - b^2/4), where b and c are real constants.