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To understand the key features of the function [tex]\( f(x) = \frac{5}{4} \sin(x) + 1 \)[/tex], we need to analyze its behavior and characteristics.
1. Maximum Value:
- The function [tex]\( f(x) = \frac{5}{4} \sin(x) + 1 \)[/tex] reaches its maximum when [tex]\(\sin(x)\)[/tex] is at its maximum value, which is 1.
- Substituting [tex]\(\sin(x) = 1\)[/tex] into the function, we get:
[tex]\[ f(x)_{\text{max}} = \frac{5}{4} \cdot 1 + 1 = \frac{5}{4} + 1 = \frac{9}{4} = 2.25 \][/tex]
- Therefore, the maximum value of the function is [tex]\(2.25\)[/tex].
2. Minimum Value:
- The function [tex]\( f(x) = \frac{5}{4} \sin(x) + 1 \)[/tex] reaches its minimum when [tex]\(\sin(x)\)[/tex] is at its minimum value, which is -1.
- Substituting [tex]\(\sin(x) = -1\)[/tex] into the function, we get:
[tex]\[ f(x)_{\text{min}} = \frac{5}{4} \cdot (-1) + 1 = -\frac{5}{4} + 1 = -\frac{5}{4} + \frac{4}{4} = -\frac{1}{4} = -0.25 \][/tex]
- Therefore, the minimum value of the function is [tex]\(-0.25\)[/tex].
3. Behavior on the Interval [tex]\((0, \frac{\pi}{2})\)[/tex]:
- To determine the behavior of the function on the interval [tex]\((0, \frac{\pi}{2})\)[/tex], we examine the behavior of the sine function in this interval.
- On the interval [tex]\((0, \frac{\pi}{2})\)[/tex], [tex]\(\sin(x)\)[/tex] increases from 0 to 1. Since [tex]\( \frac{5}{4} \sin(x) + 1 \)[/tex] is a linear transformation of [tex]\(\sin(x)\)[/tex], it retains this increasing behavior.
- Therefore, the function [tex]\( f(x) \)[/tex] is increasing on the interval [tex]\((0, \frac{\pi}{2})\)[/tex].
4. Range of the Function:
- The range of the function is determined by its minimum and maximum values.
- As previously determined, the minimum value is [tex]\(-0.25\)[/tex] and the maximum value is [tex]\(2.25\)[/tex].
- Therefore, the range of the function is from [tex]\(-0.25\)[/tex] to [tex]\(2.25\)[/tex].
In conclusion:
- The maximum value of the function is [tex]\( \boxed{2.25} \)[/tex]
- The minimum value of the function is [tex]\( \boxed{-0.25} \)[/tex]
- On the interval [tex]\( \left(0, \frac{\pi}{2}\right) \)[/tex], the function is [tex]\( \boxed{\text{increasing}} \)[/tex]
- The range of the function is [tex]\( \boxed{(-0.25, 2.25)} \)[/tex]
1. Maximum Value:
- The function [tex]\( f(x) = \frac{5}{4} \sin(x) + 1 \)[/tex] reaches its maximum when [tex]\(\sin(x)\)[/tex] is at its maximum value, which is 1.
- Substituting [tex]\(\sin(x) = 1\)[/tex] into the function, we get:
[tex]\[ f(x)_{\text{max}} = \frac{5}{4} \cdot 1 + 1 = \frac{5}{4} + 1 = \frac{9}{4} = 2.25 \][/tex]
- Therefore, the maximum value of the function is [tex]\(2.25\)[/tex].
2. Minimum Value:
- The function [tex]\( f(x) = \frac{5}{4} \sin(x) + 1 \)[/tex] reaches its minimum when [tex]\(\sin(x)\)[/tex] is at its minimum value, which is -1.
- Substituting [tex]\(\sin(x) = -1\)[/tex] into the function, we get:
[tex]\[ f(x)_{\text{min}} = \frac{5}{4} \cdot (-1) + 1 = -\frac{5}{4} + 1 = -\frac{5}{4} + \frac{4}{4} = -\frac{1}{4} = -0.25 \][/tex]
- Therefore, the minimum value of the function is [tex]\(-0.25\)[/tex].
3. Behavior on the Interval [tex]\((0, \frac{\pi}{2})\)[/tex]:
- To determine the behavior of the function on the interval [tex]\((0, \frac{\pi}{2})\)[/tex], we examine the behavior of the sine function in this interval.
- On the interval [tex]\((0, \frac{\pi}{2})\)[/tex], [tex]\(\sin(x)\)[/tex] increases from 0 to 1. Since [tex]\( \frac{5}{4} \sin(x) + 1 \)[/tex] is a linear transformation of [tex]\(\sin(x)\)[/tex], it retains this increasing behavior.
- Therefore, the function [tex]\( f(x) \)[/tex] is increasing on the interval [tex]\((0, \frac{\pi}{2})\)[/tex].
4. Range of the Function:
- The range of the function is determined by its minimum and maximum values.
- As previously determined, the minimum value is [tex]\(-0.25\)[/tex] and the maximum value is [tex]\(2.25\)[/tex].
- Therefore, the range of the function is from [tex]\(-0.25\)[/tex] to [tex]\(2.25\)[/tex].
In conclusion:
- The maximum value of the function is [tex]\( \boxed{2.25} \)[/tex]
- The minimum value of the function is [tex]\( \boxed{-0.25} \)[/tex]
- On the interval [tex]\( \left(0, \frac{\pi}{2}\right) \)[/tex], the function is [tex]\( \boxed{\text{increasing}} \)[/tex]
- The range of the function is [tex]\( \boxed{(-0.25, 2.25)} \)[/tex]
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