Find the absolute maximum and absolute minimum values of f on the given interval.
f(x) = xe^((−x^2)/98), [−6, 14]
To find the absolute maximum and absolute minimum values of the function f(x) = xe^((-x^2)/98) on the interval [−6, 14], we need to find the critical points and check the function values at those points and the endpoints of the interval.
First, let's find the critical points by finding where the derivative of f(x) is equal to zero or undefined.
1. Find the derivative of f(x):
f'(x) = e^((-x^2)/98) - (2x^2)e^((-x^2)/98)/98
2. Set f'(x) equal to zero and solve for x:
e^((-x^2)/98) - (2x^2)e^((-x^2)/98)/98 = 0
Factoring out e^((-x^2)/98), we get:
e^((-x^2)/98)(1 - (2x^2)/98) = 0
Setting each factor equal to zero gives us two possible critical points:
e^((-x^2)/98) = 0. This is not possible since e^((-x^2)/98) is always positive.
1 - (2x^2)/98 = 0
Solving for x, we get:
(2x^2)/98 = 1
2x^2 = 98
x^2 = 49
x = ±7
So, the two critical points on the interval [−6, 14] are x = -7 and x = 7.
Next, we need to check the function values at these critical points and the endpoints of the interval.
1. Endpoint: x = -6
Evaluate f(-6):
f(-6) = (-6)e^((-(-6)^2)/98) = -6e^((-36)/98)
2. Endpoint: x = 14
Evaluate f(14):
f(14) = (14)e^((-14^2)/98) = 14e^((-196)/98)
3. Critical point: x = -7
Evaluate f(-7):
f(-7) = (-7)e^((-(-7)^2)/98) = -7e^((-49)/98)
4. Critical point: x = 7
Evaluate f(7):
f(7) = (7)e^((-7^2)/98) = 7e^((-49)/98)
Now, compare the function values to determine the absolute maximum and minimum:
-6e^((-36)/98) ≈ -5.560 (at x = -6)
14e^((-196)/98) ≈ 0.198 (at x = 14)
-7e^((-49)/98) ≈ -5.715 (at x = -7)
7e^((-49)/98) ≈ 0.134 (at x = 7)
From the comparison, we find:
Absolute maximum value ≈ 0.198 at x = 14
Absolute minimum value ≈ -5.715 at x = -7
To find the absolute maximum and minimum values of a function on a given interval, we need to follow these steps:
Step 1: Find the critical points of the function within the given interval by taking the derivative and setting it equal to zero.
Step 2: Evaluate the function at the critical points and the endpoints of the interval.
Step 3: Compare the values obtained in step 2 to determine the absolute maximum and minimum values.
Let's begin with your function f(x) = xe^((-x^2)/98), defined on the interval [-6, 14].
Step 1: Find the critical points.
To find the critical points, we need to take the derivative of f(x) with respect to x and set it equal to zero.
f'(x) = e^((-x^2)/98) - (2x^2/98)e^((-x^2)/98)
Setting f'(x) = 0, we have:
e^((-x^2)/98) - (2x^2/98)e^((-x^2)/98) = 0
Factoring out e^((-x^2)/98), we get:
e^((-x^2)/98)[1 - (2x^2/98)] = 0
This equation is satisfied when either e^((-x^2)/98) = 0 (which is not possible), or when 1 - (2x^2/98) = 0.
Simplifying the second equation, we have:
1 - (2x^2/98) = 0
2x^2 = 98
x^2 = 49
x = ±7
So the critical points are x = -7 and x = 7.
Step 2: Evaluate the function at the critical points and endpoints.
We need to evaluate f(x) at x = -6, x = 7, and x = 14, as well as the critical points x = -7 and x = 7.
f(-6) = -6e^((-6^2)/98) ≈ -5.970
f(7) = 7e^((7^2)/98) ≈ 3.141
f(14) = 14e^((14^2)/98) ≈ 8.412
f(-7) = -7e^((-7^2)/98) ≈ -6.467
f(7) = 7e^((7^2)/98) ≈ 3.141
Step 3: Compare the values obtained.
By comparing the values obtained in step 2, we can determine the absolute maximum and minimum values.
The absolute maximum value is 8.412, which occurs at x = 14.
The absolute minimum value is -6.467, which occurs at x = -7.
Therefore, the absolute maximum and minimum values of f(x) = xe^((-x^2)/98) on the interval [-6, 14] are 8.412 and -6.467, respectively.
f ' (x) = x( xe^((−x^2)/98)(-2x/98) + e^((−x^2)/98)
= e^((−x^2)/98) ( -x^2/49) + 1)
= 0 for a max/min
-x^2/49 +1 = 0
x^2 = 49
x = ± 7
evaluate
f(7)
f(-7)
f(-6)
f(14)
and determine which is the largest and which is the smallest