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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].
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