Find all points on the graph of the function 
f(x) = 2 sin(x) + sin2(x)
 at which the tangent line is horizontal. (Use n as your arbitrary integer.)
(x, y)
=	

 
  
Incorrect: Your answer is incorrect.
 

 (smaller y-value)
(x, y)
=	

 
  
Incorrect: Your answer is incorrect.
 

 (larger y-value)

To find all points on the graph of the function \( f(x) = 2\sin(x) + \sin^2(x) \) at which the tangent line is horizontal, we need to follow these steps:

### Step 1: Find the derivative of the function
The tangent line is horizontal when the derivative of the function is zero. Let's differentiate \( f(x) = 2\sin(x) + \sin^2(x) \) with respect to \( x \).

\[
f'(x) = \frac{d}{dx}\left( 2\sin(x) \right) + \frac{d}{dx}\left( \sin^2(x) \right)
\]

Using the chain rule for \( \sin^2(x) \), we get:

\[
f'(x) = 2\cos(x) + 2\sin(x)\cos(x)
\]

Simplifying:

\[
f'(x) = 2\cos(x)(1 + \sin(x))
\]

### Step 2: Set the derivative equal to zero
We now set the derivative equal to zero to find the points where the tangent line is horizontal:

\[
2\cos(x)(1 + \sin(x)) = 0
\]

This equation is satisfied if either factor is zero:

1. \( \cos(x) = 0 \)
2. \( 1 + \sin(x) = 0 \)

#### Case 1: \( \cos(x) = 0 \)

The cosine function equals zero at:

\[
x = \frac{\pi}{2} + n\pi \quad \text{for any integer } n
\]

#### Case 2: \( 1 + \sin(x) = 0 \)

Solving for \( \sin(x) \), we get:

\[
\sin(x) = -1
\]

The sine function equals -1 at:

\[
x = \frac{3\pi}{2} + 2n\pi \quad \text{for any integer } n
\]

### Step 3: Find the corresponding \( y \)-values
We now substitute the \( x \)-values back into the original function \( f(x) = 2\sin(x) + \sin^2(x) \) to find the corresponding \( y \)-values.

#### For \( x = \frac{\pi}{2} + n\pi \):
At \( x = \frac{\pi}{2} \), \( \sin\left(\frac{\pi}{2}\right) = 1 \):

\[
f\left(\frac{\pi}{2}\right) = 2(1) + (1)^2 = 2 + 1 = 3
\]

At \( x = \frac{3\pi}{2} \), \( \sin\left(\frac{3\pi}{2}\right) = -1 \):

\[
f\left(\frac{3\pi}{2}\right) = 2(-1) + (-1)^2 = -2 + 1 = -1
\]

Thus, the points where the tangent line is horizontal are:

\[
(x, y) = \left(\frac{\pi}{2} + n\pi, 3\right) \quad \text{(larger \( y \)-value)}
\]
\[
(x, y) = \left(\frac{3\pi}{2} + 2n\pi, -1\right) \quad \text{(smaller \( y \)-value)}
\]

### Final Answer:
1. \( (x, y) = \left(\frac{\pi}{2}, 3\right) \) (smaller \( y \)-value)
2. \( (x, y) = \left(\frac{3\pi}{2}, -1\right) \) (larger \( y \)-value)

Let me know if you would like to discuss the steps in more detail.

---

Here are 5 related questions to explore:
1. How would the process change if we had \( f(x) = \sin(x) + \sin^2(x) \) instead?
2. What is the geometric interpretation of a horizontal tangent line?
3. Can you find points where the tangent line has a slope of 1?
4. How would the second derivative help in confirming the type of extremum?
5. What if we had \( f(x) = 2\sin(x) + \cos^2(x) \)? Would the solution be similar?

**Tip**: Always check critical points by plugging them back into the original equation to ensure correctness.

Find all points on the graph of the function f(x) = 2sin(x) + sin^2(x) at which the tangent line is horizontal. (Use n as your arbitrary integer.)

Discover how to find all points on the graph of f(x) = 2sin(x) + sin^2(x) where the tangent line is horizontal. This problem involves taking the derivative, setting it to zero, and solving using trigonometric identities. Suitable for advanced high school or undergraduate students.

Math Problem Statement

Find all points on the graph of the function f(x) = 2 sin(x) + sin2(x) at which the tangent line is horizontal. (Use n as your arbitrary integer.) (x, y)

(smaller y-value) (x, y)

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