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 of the maximum of the absolute value of f(x) over the interval [-10, 10], we need to consider two cases:
1. The maximum occurs within the interval [-10, 10].
2. The maximum occurs at one of the endpoints (-10 or 10).
Let's start with case 1:
We can find the maximum value within the interval by finding the critical points of f(x) and then evaluating the function at those points.
To find the critical points, we need to find the values of x where the derivative of f(x) is equal to zero.
Step 1: Take the derivative of f(x):
f'(x) = 2x + b
Step 2: Set f'(x) equal to zero and solve for x:
2x + b = 0
x = -b/2
Now, let's evaluate f(x) at the critical point x = -b/2:
f(-b/2) = (-b/2)^2 + b(-b/2) + c
= b^2/4 - b^2/2 + c
= (c - b^2/4)
So, the maximum value of |f(x)| within the interval [-10, 10] occurs at x = -b/2, and its value is |c - b^2/4|.
Now, let's move on to case 2:
We need to evaluate f(x) at the endpoints of the interval [-10, 10].
f(-10) = (-10)^2 + b(-10) + c
= 100 - 10b + c
f(10) = (10)^2 + b(10) + c
= 100 + 10b + c
The maximum value of |f(x)| at the endpoints is the larger value between |100 - 10b + c| and |100 + 10b + c|.
To find the minimum of the maximum of |f(x)|, we need to find the minimum value among the maximum values from case 1 and case 2.
So, the minimum of the maximum of |f(x)| over the interval [-10, 10] can be found as follows:
1. Evaluate |c - b^2/4|, which is the maximum value within the interval (-b/2).
2. Evaluate the larger value between |100 - 10b + c| and |100 + 10b + c|, which accounts for the maximum at the endpoints (-10 and 10).
3. Take the minimum value among the results in steps 1 and 2.
Note: To find the exact minimum value, you would need specific values for b and c.