Find the absolute extrema of the function. (Round your answer to three decimal places.)
f(x) = xe^-x2 on [0,2]
Absolute maximum value:
at x =
Absolute minimum value:
at x =
See:
http://www.jiskha.com/display.cgi?id=1373656357
To find the absolute extrema of a function on an interval, we need to evaluate the function at the critical points and endpoints of the interval.
First, let's find the critical points by taking the derivative of the function f(x) with respect to x and setting it equal to zero:
f(x) = xe^(-x^2)
To find the derivative, we can use the product rule:
f'(x) = e^(-x^2) - 2x^2e^(-x^2)
Setting f'(x) equal to zero and solving for x, we have:
e^(-x^2) - 2x^2e^(-x^2) = 0
e^(-x^2)(1 - 2x^2) = 0
So, either e^(-x^2) = 0 or 1 - 2x^2 = 0.
Since e^(-x^2) is always positive, it can never be equal to zero. Therefore, we only need to solve the equation 1 - 2x^2 = 0:
2x^2 = 1
x^2 = 1/2
x = ±√(1/2)
Since the interval is [0, 2], we only need to consider the critical point x = √(1/2) because it lies within the interval.
Next, we evaluate the function at the critical point and the endpoints of the interval:
f(0) = 0e^(-0^2) = 0
f(√(1/2)) = √(1/2)e^(-(√(1/2))^2) = √(1/2)e^(-1/2) ≈ 0.483
f(2) = 2e^(-2^2) = 2e^(-4) ≈ 0.036
We can see that f(0) = 0 is the absolute minimum value because it is the smallest value of the function within the interval [0, 2].
For the absolute maximum value, we compare the values of f(√(1/2)) and f(2), and we can see that f(√(1/2)) ≈ 0.483 is the larger value. Hence, the absolute maximum value is approximately 0.483, which occurs at x = √(1/2).
Therefore, the absolute minimum value is 0 at x = 0, and the absolute maximum value is approximately 0.483 at x = √(1/2).