Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To solve the given problem, we need to determine the minimum and maximum values of the function [tex]\( f(x) = \sin^{-1}(2x) + \sin(2x) + \cos^{-1}(2x) + \cos(2x) \)[/tex] within its domain, and then find the sum of these values. Let's break down the steps:
1. Understanding the Function and its Domain:
- The function involved contains [tex]\(\sin^{-1}(2x)\)[/tex] and [tex]\(\cos^{-1}(2x)\)[/tex], which have restricted domains.
- For the function [tex]\(\sin^{-1}(x)\)[/tex], it is defined when [tex]\( -1 \le x \le 1 \)[/tex].
- However, because we're dealing with [tex]\(2x\)[/tex], we need [tex]\( -1 \le 2x \le 1 \)[/tex], which simplifies to [tex]\( -\frac{1}{2} \le x \le \frac{1}{2} \)[/tex].
2. Finding Extremes of the Function:
- We evaluate the function [tex]\( f(x) \)[/tex] at the endpoints of the domain [tex]\( -\frac{1}{2} \)[/tex] and [tex]\( \frac{1}{2} \)[/tex].
3. Evaluations at Boundary Points:
- For [tex]\( x = -\frac{1}{2} \)[/tex]:
[tex]\[ f\left(-\frac{1}{2}\right) = \sin^{-1}\left(-1\right) + \sin\left(-1\right) + \cos^{-1}\left(-1\right) + \cos\left(-1\right) \][/tex]
Simplifying, we get:
[tex]\[ \sin^{-1}(-1) = -\frac{\pi}{2}, \quad \cos^{-1}(-1) = \pi, \quad \sin(-1) \approx -0.8415, \quad \cos(-1) \approx 0.5403 \][/tex]
Therefore:
[tex]\[ f\left(-\frac{1}{2}\right) = -\frac{\pi}{2} + (-0.8415) + \pi + 0.5403 \approx 1.2696 \][/tex]
- For [tex]\( x = \frac{1}{2} \)[/tex]:
[tex]\[ f\left(\frac{1}{2}\right) = \sin^{-1}(1) + \sin(1) + \cos^{-1}(1) + \cos(1) \][/tex]
Simplifying, we get:
[tex]\[ \sin^{-1}(1) = \frac{\pi}{2}, \quad \cos^{-1}(1) = 0, \quad \sin(1) \approx 0.8415, \quad \cos(1) \approx 0.5403 \][/tex]
Therefore:
[tex]\[ f\left(\frac{1}{2}\right) = \frac{\pi}{2} + 0.8415 + 0 + 0.5403 \approx 2.9526 \][/tex]
4. Determining Minimum and Maximum:
- The values we found are [tex]\( f\left(-\frac{1}{2}\right) \approx 1.2696 \)[/tex] and [tex]\( f\left(\frac{1}{2}\right) \approx 2.9526 \)[/tex].
- So, the minimum value [tex]\( m \)[/tex] is approximately [tex]\( 1.2696 \)[/tex] and the maximum value [tex]\( M \)[/tex] is approximately [tex]\( 2.9526 \)[/tex].
5. Sum of Minimum and Maximum Values:
- Adding these, we get [tex]\( m + M \)[/tex]:
[tex]\[ m + M \approx 1.2696 + 2.9526 = 4.2222 \][/tex]
Hence, the sum of the minimum and maximum values of the function [tex]\( f(x) \)[/tex] within the given domain is [tex]\( \boxed{4.2222} \)[/tex].
1. Understanding the Function and its Domain:
- The function involved contains [tex]\(\sin^{-1}(2x)\)[/tex] and [tex]\(\cos^{-1}(2x)\)[/tex], which have restricted domains.
- For the function [tex]\(\sin^{-1}(x)\)[/tex], it is defined when [tex]\( -1 \le x \le 1 \)[/tex].
- However, because we're dealing with [tex]\(2x\)[/tex], we need [tex]\( -1 \le 2x \le 1 \)[/tex], which simplifies to [tex]\( -\frac{1}{2} \le x \le \frac{1}{2} \)[/tex].
2. Finding Extremes of the Function:
- We evaluate the function [tex]\( f(x) \)[/tex] at the endpoints of the domain [tex]\( -\frac{1}{2} \)[/tex] and [tex]\( \frac{1}{2} \)[/tex].
3. Evaluations at Boundary Points:
- For [tex]\( x = -\frac{1}{2} \)[/tex]:
[tex]\[ f\left(-\frac{1}{2}\right) = \sin^{-1}\left(-1\right) + \sin\left(-1\right) + \cos^{-1}\left(-1\right) + \cos\left(-1\right) \][/tex]
Simplifying, we get:
[tex]\[ \sin^{-1}(-1) = -\frac{\pi}{2}, \quad \cos^{-1}(-1) = \pi, \quad \sin(-1) \approx -0.8415, \quad \cos(-1) \approx 0.5403 \][/tex]
Therefore:
[tex]\[ f\left(-\frac{1}{2}\right) = -\frac{\pi}{2} + (-0.8415) + \pi + 0.5403 \approx 1.2696 \][/tex]
- For [tex]\( x = \frac{1}{2} \)[/tex]:
[tex]\[ f\left(\frac{1}{2}\right) = \sin^{-1}(1) + \sin(1) + \cos^{-1}(1) + \cos(1) \][/tex]
Simplifying, we get:
[tex]\[ \sin^{-1}(1) = \frac{\pi}{2}, \quad \cos^{-1}(1) = 0, \quad \sin(1) \approx 0.8415, \quad \cos(1) \approx 0.5403 \][/tex]
Therefore:
[tex]\[ f\left(\frac{1}{2}\right) = \frac{\pi}{2} + 0.8415 + 0 + 0.5403 \approx 2.9526 \][/tex]
4. Determining Minimum and Maximum:
- The values we found are [tex]\( f\left(-\frac{1}{2}\right) \approx 1.2696 \)[/tex] and [tex]\( f\left(\frac{1}{2}\right) \approx 2.9526 \)[/tex].
- So, the minimum value [tex]\( m \)[/tex] is approximately [tex]\( 1.2696 \)[/tex] and the maximum value [tex]\( M \)[/tex] is approximately [tex]\( 2.9526 \)[/tex].
5. Sum of Minimum and Maximum Values:
- Adding these, we get [tex]\( m + M \)[/tex]:
[tex]\[ m + M \approx 1.2696 + 2.9526 = 4.2222 \][/tex]
Hence, the sum of the minimum and maximum values of the function [tex]\( f(x) \)[/tex] within the given domain is [tex]\( \boxed{4.2222} \)[/tex].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.