(2 points) Consider the function f(x)=xe^−8x, 0≤x≤2.
This function has an absolute minimum value equal to:
which is attained at x=
and an absolute maximum value equal to:
which is attained at x=
max/min occur when f'(x) = 0
Assuming you meant f(x) = xe^(-8x), then
f'(x) = (1-8x) e^(-8x)
So, there will be a local min/max (which?) at x = 1/8
Now check f(0) and f(2) to see whether they are global min/max.
Well, let's find the absolute minimum and maximum values for this function, shall we? But first, let's make this a fun exercise!
*fumbles with clown hat*
Alright, let's put on our math clown shoes and grab a clown calculator!
*fumbles with calculator*
Now, we want to find the absolute minimum and maximum values of the function f(x) = xe^(-8x) when 0 ≤ x ≤ 2. It sounds like we're going on a wild mathematical ride!
*fills balloon with anticipation*
To find the minimum and maximum values, we need to find the critical points. So, let's put on our detective clown hats and find where the derivative of the function is equal to zero or undefined!
*fumbles with magnifying glass*
Taking the derivative of f(x) = xe^(-8x), we get f'(x) = e^(-8x) - 8xe^(-8x). Let's set this equal to zero!
*eagerly crunches numbers*
Solving for x, we find x = 1/8. Ah, how wonderfully symmetric! But wait, we have to check if this point is within our given interval.
*fumbles with calendar*
Ah, yes! The interval is 0 ≤ x ≤ 2. Luckily, 1/8 falls within this range! So, x = 1/8 is a critical point!
*fumbles with a clown cape*
Now, let's check the endpoints of our interval, shall we?
*fumbles with a compass*
At x = 0, the value of f(x) is 0. So, we have a potential minimum there!
*fumbles with a mini trampoline*
And at x = 2, the value of f(x) is 2e^(-16), which is a fantastically small number!
*fumbles with a confetti cannon*
So, putting it all together, we have the following:
The absolute minimum value of f(x) is 0, attained at x = 0!
And the absolute maximum value of f(x) is 2e^(-16), attained at x = 2!
*takes a bow with a clown horn honk*
To find the absolute minimum and maximum values of the function f(x) = xe^(-8x), we need to first find the critical points by taking the derivative of the function and setting it equal to zero.
Let's start by finding the derivative of f(x):
f'(x) = e^(-8x) - 8xe^(-8x)
Setting f'(x) equal to zero:
e^(-8x) - 8xe^(-8x) = 0
Factoring out e^(-8x):
e^(-8x)(1 - 8x) = 0
Setting each factor equal to zero:
e^(-8x) = 0 --> This is not possible since exponential functions are always positive.
1 - 8x = 0
Solving for x:
-8x = -1
x = 1/8
Now, we need to check the endpoints of the interval [0, 2] as well as the critical point x = 1/8 for the absolute minimum and maximum values of f(x).
Checking the endpoint x = 0:
f(0) = 0 * e^(-8 * 0) = 0 * e^0 = 0
Checking the endpoint x = 2:
f(2) = 2 * e^(-8 * 2) = 2 * e^(-16)
To determine if the critical point x = 1/8 is the absolute minimum or maximum, we can compare the values of f(0), f(2), and f(1/8).
f(0) = 0
f(2) = 2 * e^(-16) ≈ 0.000457
f(1/8) = (1/8) * e^(-8 * 1/8) ≈ 0.036538
Therefore, the absolute minimum value of f(x) is 0, attained at x = 0, and the absolute maximum value is approximately 0.000457, attained at x = 2.
To find the absolute minimum and maximum values of a function, we can use the concept of critical points and the endpoints of the given interval.
Step 1: Find the critical points by setting the derivative of the function equal to zero and solving for x.
The derivative of f(x) = xe^(-8x) can be found using the product rule:
f'(x) = e^(-8x) + xe^(-8x)(-8)
Setting this equal to zero, we have:
e^(-8x) - 8xe^(-8x) = 0
Factoring out e^(-8x), we get:
e^(-8x)(1 - 8x) = 0
Now, set each term equal to zero:
e^(-8x) = 0 (No solution)
1 - 8x = 0
From this, we find that x = 1/8.
Step 2: Find the function values at the critical points and the endpoints.
We need to evaluate the function f(x) at the critical points (x = 1/8) and the endpoints (x = 0 and x = 2).
f(0) = 0e^0 = 0
f(1/8) = (1/8)e^(-8/8) = (1/8)e^(-1) = 1/8e^(-1)
f(2) = 2e^(-8(2)) = 2e^(-16)
Step 3: Identify the absolute minimum and maximum values.
To find the absolute minimum value, compare the function values at the critical points and endpoints. The smallest value is the absolute minimum.
The smallest value among f(0), f(1/8), and f(2) will be the absolute minimum.
To find the absolute maximum value, look for the largest value among f(0), f(1/8), and f(2). This will be the absolute maximum.
Comparing these values, we can determine the absolute minimum and maximum:
Absolute minimum value:
Absolute maximum value:
At x= 0.
At x= 2.