What is the maximum value of the function `f(x) = 4 sin^(2)x + 1`?

Options

A. 5
B. 3
C. 2
D. 1

Answer

To find the maximum value of the function \( f(x) = 4 \sin^2 x + 1 \), we can follow these steps:

### Step 1: Understand the range of \( \sin^2 x \)
The sine function, \( \sin x \), has a range of \([-1, 1]\). When we square this function, the range changes.

- The minimum value of \( \sin^2 x \) occurs when \( \sin x = 0 \), giving \( \sin^2 x = 0 \).
- The maximum value of \( \sin^2 x \) occurs when \( \sin x = 1 \) or \( \sin x = -1 \), giving \( \sin^2 x = 1 \).

Thus, the range of \( \sin^2 x \) is \([0, 1]\).

### Step 2: Substitute the range into the function
Now we substitute the range of \( \sin^2 x \) into the function \( f(x) \):

- When \( \sin^2 x = 0 \):
\[ f(x) = 4(0) + 1 = 1 \]

- When \( \sin^2 x = 1 \):
\[ f(x) = 4(1) + 1 = 4 + 1 = 5 \]

### Step 3: Determine the maximum value
From the calculations above, we see that:
- The minimum value of \( f(x) \) is \( 1 \).
- The maximum value of \( f(x) \) is \( 5 \).

### Conclusion
Thus, the maximum value of the function \( f(x) = 4 \sin^2 x + 1 \) is \( 5 \).