Let's analyze the function [tex]\( f(x) = \frac{5}{4} \sin(x) + 1 \)[/tex].
1. Maximum Value:
The sine function, [tex]\( \sin(x) \)[/tex], has a maximum value of 1. Substituting this into the function, we get:
[tex]\[
f(x) = \frac{5}{4} \times 1 + 1 = \frac{5}{4} + 1 = 2.25
\][/tex]
Therefore, the maximum value of the function is [tex]\( 2.25 \)[/tex].
2. Minimum Value:
The sine function, [tex]\( \sin(x) \)[/tex], has a minimum value of -1. Substituting this into the function, we get:
[tex]\[
f(x) = \frac{5}{4} \times (-1) + 1 = -\frac{5}{4} + 1 = -1.25 + 1 = -0.25
\][/tex]
Therefore, the minimum value of the function is [tex]\( -0.25 \)[/tex].
3. Behavior 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] is strictly increasing from 0 to 1. Therefore, the function [tex]\( f(x) = \frac{5}{4} \sin(x) + 1 \)[/tex] increases on this interval as well.
4. Range:
Since the maximum value of the function is [tex]\( 2.25 \)[/tex] and the minimum value is [tex]\( -0.25 \)[/tex], the range of the function is [tex]\([-0.25, 2.25]\)[/tex].
Summarizing the key features:
- The maximum value of the function is [tex]\( 2.25 \)[/tex].
- The minimum value of the function is [tex]\( -0.25 \)[/tex].
- On the interval [tex]\(\left(0, \frac{\pi}{2}\right)\)[/tex], the function is increasing.
- The range of the function is [tex]\([-0.25, 2.25]\)[/tex].
