Alright, let's analyze the function [tex]\( f(x) = \frac{5}{4} \sin(x) + 1 \)[/tex] to determine its key features.
1. Maximum value of the function:
The sine function [tex]\(\sin(x)\)[/tex] has a maximum value of 1. Multiplying this by [tex]\(\frac{5}{4}\)[/tex] and then adding 1 gives us the maximum value:
[tex]\[
\frac{5}{4} \cdot 1 + 1 = \frac{5}{4} + 1 = \frac{5}{4} + \frac{4}{4} = \frac{9}{4} = 2.25
\][/tex]
So, the maximum value of the function is [tex]\(\boxed{2.25}\)[/tex].
2. Minimum value of the function:
The sine function [tex]\(\sin(x)\)[/tex] has a minimum value of -1. Multiplying this by [tex]\(\frac{5}{4}\)[/tex] and then adding 1 gives us the minimum value:
[tex]\[
\frac{5}{4} \cdot (-1) + 1 = -\frac{5}{4} + 1 = -\frac{5}{4} + \frac{4}{4} = -\frac{5}{4} + \frac{4}{4} = -\frac{1}{4} = -0.25
\][/tex]
So, the minimum value of the function is [tex]\(\boxed{-0.25}\)[/tex].
3. Behavior of the function on the interval [tex]\(\left(0, \frac{\pi}{2}\right)\)[/tex]:
On the interval [tex]\(\left(0, \frac{\pi}{2}\right)\)[/tex], the sine function [tex]\(\sin(x)\)[/tex] increases from 0 to 1. Therefore, since [tex]\(\sin(x)\)[/tex] is increasing, [tex]\( f(x) = \frac{5}{4} \sin(x) + 1 \)[/tex] is also increasing on this interval.
So, on the interval [tex]\(\left(0, \frac{\pi}{2}\right)\)[/tex], the function is [tex]\(\boxed{\text{increasing}}\)[/tex].
4. Range of the function:
Combining the maximum and minimum values, the range of the function [tex]\( f(x) \)[/tex] is from -0.25 to 2.25. Therefore, the range can be written as:
[tex]\[
\boxed{(-0.25, 2.25)}
\][/tex]
Summarizing, the key features are:
- 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].
