Consider the function with rule f(x) = |x^2-ax|, where a is a positive constant.
Find the maximum value of the function in the interval [0,a]
How do you do find that?
if x^2-ax >= 0, |x^2-ax| = x^2-ax
That is, if x(x-a) >= 0
x <= 0 or x >= a
These intervals are outside the domain.
If x^2-ax < 0, then |x^2-ax| = -(x^2-ax)
That is, if x(x-a) < 0
So, for 0<=x<=a the maximum value of f(x) is at x = a/2
See a sample graph here:
http://www.wolframalpha.com/input/?i=%7Cx(x-2)%7C
To find the maximum value of the function f(x) in the interval [0,a], we need to determine the critical points of the function within that interval.
1. Start by differentiating the function f(x) with respect to x.
f(x) = |x^2 - ax|
To simplify the differentiation process, we can rewrite the absolute value using piecewise notation:
f(x) = (x^2 - ax) if x^2 - ax >= 0
f(x) = -(x^2 - ax) if x^2 - ax < 0
2. Determine the critical points by setting the derivative equal to zero.
We need to solve f'(x) = 0 separately for the cases when x^2 - ax is positive and negative.
Case 1: When x^2 - ax >= 0:
f'(x) = 2x - a = 0
2x = a
x = a/2
Case 2: When x^2 - ax < 0:
f'(x) = -2x + a = 0
2x = a
x = a/2
Since the critical point, a/2, is the same for both cases, it is a potential maximum value.
3. Evaluate the function at the critical point and boundaries of the interval.
f(a/2) = |(a/2)^2 - a(a/2)|
= |(a^2/4) - (a^2/2)|
= |(-a^2/4)|
= a^2/4
Since a is a positive constant, a^2/4 will also be positive.
4. Compare the function values at the critical point and boundaries.
We know that f(a/2) = a^2/4 is positive, so we need to compare it with the function values at the interval boundaries, 0 and a.
f(0) = |0^2 - a*0| = 0
f(a) = |a^2 - a*a| = |0| = 0
Since a^2/4 is positive and greater than zero, the maximum value of the function in the interval [0,a] is a^2/4.
To find the maximum value of the function f(x) = |x^2-ax| in the interval [0, a], we need to examine the critical points and the endpoints of the interval.
1. Consider the critical points:
- Critical points occur where the derivative of the function is equal to zero or does not exist.
- The function f(x) = |x^2 - ax| is continuous and differentiable everywhere except possibly at x = 0 and x = a.
- Therefore, we need to check the critical points at x = 0 and x = a.
2. Evaluate the function at the critical points and endpoints:
- Calculate the values of f(x) at x = 0, x = a, and the values of f(x) where the derivative is equal to zero.
- Substitute these values into the function f(x) = |x^2 - ax| and find the corresponding function values.
3. Determine the maximum value:
- Compare the function values obtained in step 2 to identify the maximum value of f(x) within the interval [0, a].
- The largest value among the function values will be the maximum value of f(x).
Remember that the maximum value may occur at one of the critical points or endpoints.
By following these steps, you can find the maximum value of the function in the interval [0, a].