Find the absolute maximum and the absolute minimum values of
f(x)=x−7cos(x) on the interval [−ð,ð).
Think of
f(x)=f1(x)+f2(x)
where
f1(x)=x
f2(x)=-cos(x)
We note that f1(x) is monotonically increasing throughout [-π, π) while
f2(x) is increasing between [-π, 0) and decreasing between [0,π).
Therefore to find the absolute maximum and minimum, we only need to check three locations, x=-π, x=0, and x=π.
f(-π)=-π+7 = 3.86 (approx.)
f(0)=0-7=0-7 = -7
f(π)=π+7 = 10.14 (approx.)
from which you can pick out the absolute maximum and minimum.
To find the absolute maximum and minimum values of a function, we need to follow these steps:
1. Find the critical points of the function within the given interval.
2. Evaluate the function at the critical points.
3. Evaluate the function at the endpoints of the interval.
4. Compare all the values obtained in steps 2 and 3 and determine the absolute maximum and minimum.
Let's begin with step 1:
1. Find the critical points of f(x):
- To find the critical points, we need to find the values of x where the derivative of f(x) equals zero or is undefined.
- The derivative of f(x) can be found using the product rule: f'(x) = 1 - 7(-sin(x)) = 1 + 7sin(x).
Now, set f'(x) equal to zero and solve for x:
1 + 7sin(x) = 0
7sin(x) = -1
sin(x) = -1/7
Since sin(x) can only take certain values between -1 and 1, we can see that sin(x) = -1/7 does not have solutions within the interval [−π, π).
Therefore, there are no critical points for f(x) within the interval [−π, π).
Moving on to step 2:
Since there are no critical points within the interval, we move to step 3:
2. Evaluate the function at the endpoints of the interval:
- f(-π) = -π - 7cos(-π) = -π - 7(-1) = -π + 7
- f(π) = π - 7cos(π) = π - 7(-1) = π + 7
Lastly, move to step 4:
3. Compare the values obtained in steps 2 and 3:
- f(-π) = -π + 7 ≈ 4.28
- f(π) = π + 7 ≈ 10.28
Since there are no critical points, the absolute maximum and minimum values of f(x) occur at the endpoints of the interval:
- Absolute maximum: f(π) ≈ 10.28
- Absolute minimum: f(-π) ≈ 4.28
Therefore, the absolute maximum value of f(x) on the interval [−π, π) is approximately 10.28, and the absolute minimum value is approximately 4.28.