Relative extrema of absolute functions

In summary: The critical points are x=0, 3, and 3/2.In summary, the function f(x)=|3x-x^2| has three relative extrema: local minima at x=0 and x=3, and a critical point at x=3/2. These can be found by splitting the function into two parts and using the definition of absolute value.
  • #1
John O' Meara
330
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Find the relative extrema of [tex] f(x)=|3x-x^2|[/tex].
Normally you solve f'(x)=0 to find the critical points, in that case we have 3-2x=0 => x=3/2. However there are two other relative extrema that I cannot find by calculus as the absolute nature of f(x) has confused me. Indeed doing a sign analysis on the points x=0 and 3; shows that these points are not extrema points! I am studing on my own, Please help, Thanks.
 
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  • #2
Split the function into two parts, where neither have absolute value bars,using the definition of absolute value.

Then find the critical points of each, and adjoin the point where the two meet (probably another relative extremum).
 
  • #3
Ok,
|x| = x if x > 0, -x if x < 0. |3x-x^2|= 3x-x^2 if x>3, I do not know the rest?
 
  • #4
I think I have it now, |3x-x^2|= (x-3)^2 if x>3, 3x-x^2 if 0< x <3, -x^2 if x<0
 
  • #5
John O' Meara said:
Find the relative extrema of [tex] f(x)=|3x-x^2|[/tex].
Normally you solve f'(x)=0 to find the critical points
Your first statement is incorrect. "Critical points" are points where the derivative is 0 or where the derivative does not exist.

, in that case we have 3-2x=0 => x=3/2. However there are two other relative extrema that I cannot find by calculus as the absolute nature of f(x) has confused me. Indeed doing a sign analysis on the points x=0 and 3; shows that these points are not extrema points! I am studing on my own, Please help, Thanks.

The derivative does not exist where f(x)= 0. That is, at x=0 or at x= 3. Since the value of f is 0 there and the absolute value cannot be negative, they are obviously local minima.
 

FAQ: Relative extrema of absolute functions

1. What is a relative extrema of an absolute function?

A relative extrema of an absolute function is a point on the graph where the function reaches its highest or lowest value within a specific interval. This point is relative to the surrounding points and is not necessarily the absolute highest or lowest point of the entire function.

2. How do you find the relative extrema of an absolute function?

To find the relative extrema of an absolute function, you must first take the derivative of the function. Then, set the derivative equal to zero and solve for the variable. The resulting values will be the x-coordinates of the relative extrema. To find the y-coordinate, plug the x-coordinate into the original function.

3. Can an absolute function have more than one relative extrema?

Yes, an absolute function can have multiple relative extrema. This can occur when the function has multiple peaks or valleys within a given interval. Each of these points would be considered a relative extrema.

4. What is the significance of finding relative extrema in a real-world scenario?

In a real-world scenario, finding the relative extrema can help in identifying the maximum or minimum values of a function, which can have practical applications in fields such as economics, physics, and engineering. It can also help in analyzing the behavior and trends of a function.

5. Is there a difference between relative extrema and absolute extrema?

Yes, there is a difference between relative extrema and absolute extrema. Absolute extrema are the highest or lowest points of a function over its entire domain, while relative extrema are the highest or lowest points within a specific interval. Absolute extrema are also known as global extrema, while relative extrema are known as local extrema.

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