f(x)=xe^(-2x) with domain 0< or equal to x< or equal to 10
find all values of x for which the graph of f is increasing and all values of x for which the graph is decreasing
give the x and y coordinates if all absolute max and min points
To find the values of x for which the function f(x) = xe^(-2x) is increasing or decreasing, we can start by finding the derivative of f(x).
Step 1: Find the derivative of f(x)
Using the product rule, we can find the derivative of f(x) = xe^(-2x) as follows:
f'(x) = (1)(e^(-2x)) + (x)(-2)(e^(-2x))
= e^(-2x) - 2xe^(-2x)
= e^(-2x)(1 - 2x)
Step 2: Determine where the derivative is positive or negative
To find where the graph of f(x) is increasing or decreasing, we need to determine the signs of the derivative at various intervals.
Setting f'(x) = 0, we have:
e^(-2x)(1 - 2x) = 0
This equation is satisfied when either e^(-2x) = 0 (which has no solution) or 1 - 2x = 0. Solving the latter equation for x, we find x = 1/2.
Step 3: Apply the First Derivative Test
We can now use the first derivative test to determine where the graph of f(x) is increasing or decreasing.
When x < 1/2, the factor e^(-2x) is positive while (1 - 2x) is negative. Hence, f'(x) is negative, indicating that the graph of f(x) is decreasing in this interval.
When x > 1/2, both factors e^(-2x) and (1 - 2x) are positive. Hence, f'(x) is positive, indicating that the graph of f(x) is increasing in this interval.
Step 4: Determine the end behavior of f(x)
We need to consider the behavior of f(x) as x approaches the boundaries of the given domain (0 and 10 in this case).
As x approaches 0 from the right, f(x) approaches 0.
As x approaches 10 from the left, f(x) tends towards negative infinity.
Step 5: Find the absolute maximum and minimum points
To find the absolute maximum and minimum points of f(x), we need to consider the critical points and endpoints of the given domain.
Since the function f(x) = xe^(-2x) is continuous over the closed interval [0, 10], we can use the Extreme Value Theorem.
1. Critical Points:
We found earlier that the critical point is x = 1/2.
2. Endpoints of the Domain:
At x = 0, we have f(0) = 0.
At x = 10, we have f(10) ≈ -0.1353.
To summarize:
- The graph of f(x) is decreasing on the interval [0, 1/2].
- The graph of f(x) is increasing on the interval [1/2, 10].
Absolute Extrema:
- Absolute maximum: (10, -0.1353)
- Absolute minimum: (0, 0)