Find all relative extreme points y=x e^-x
Find the definition of a relative extreme point, for example:
http://en.wikipedia.org/wiki/Extreme_point
hint:
1. find the domain of the function
f(x)=x*e^(-x)
...the domain is (-∞,∞)
2. calculate f'(x), and verify that f'(x) exists within the domain.
3. find the zeroes of f'(x)=0
4. check that the zeroes found in (3) are maxima or minima by evaluating f"(x).
If f"(x)=0, it is an inflection point and is not an extremum, thus not a critical point.
There should be one relative extreme point for the given function on its domain.
I still can't find the answer because I got max (1,1/e)
I had the impression that the question was looking for relative extreme values in calculus, in which case (1,1/e) is the answer. The function tends to -∞ as x-> -∞ and f(x)->0 as x->∞, and is a maximum at x=1 (when y=1/e).
I am sorry that the reference for extreme point in the Wiki article does not refer to the same subject.
Can you check if the following link refers to what you are working on in class?
http://en.wikibooks.org/wiki/Calculus/Extreme_Value_Theorem
To find the relative extreme points of the function y = x e^(-x), we need to determine the critical points and then evaluate the function at those points.
1. Find the first derivative of the function:
y' = e^(-x) - x e^(-x)
2. Set the derivative equal to zero to find the critical points:
e^(-x) - x e^(-x) = 0
3. Solve for x:
e^(-x) (1 - x) = 0
We have two cases:
a) e^(-x) = 0 (This case has no solution)
b) 1 - x = 0 => x = 1
Therefore, the only critical point is x = 1.
4. Determine the nature of the critical point:
To determine whether the critical point is a relative maximum or minimum, we need to analyze the second derivative.
Find the second derivative of the function:
y'' = -e^(-x) + x e^(-x)
Plug in the critical point x = 1:
y''(1) = -e^(-1) + e^(-1) = 0
Since the second derivative is zero, the test fails to provide conclusive information about the nature of the critical point.
5. Evaluate the function at the critical point and nearby points:
Plug x = 1 into the original function:
y(1) = 1 * e^(-1) = e^(-1) ≈ 0.3679
To determine whether the critical point is a relative maximum or minimum, we need to evaluate the function at points slightly greater and slightly smaller than x = 1.
For example:
y(0.9) ≈ 0.5497
y(1.1) ≈ 0.3328
Comparing these values, we find that y(0.9) > y(1) > y(1.1).
Therefore, at the critical point x = 1, there is a relative maximum.
In summary, the relative extreme point of the function y = x e^(-x) is (1, e^(-1)).