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To find the critical points of the function \( g(x) = e^x \cos x \) on the interval \([0, 2\pi]\), we need to follow these steps:

### Step 1: Differentiate \( g(x) \)

The critical points occur where the derivative of the function equals zero or does not exist. We first compute the derivative \( g'(x) \).

Using the product rule for derivatives:
\[
g'(x) = \frac{d}{dx} (e^x \cos x) = \frac{d}{dx} (e^x) \cdot \cos x + e^x \cdot \frac{d}{dx} (\cos x)
\]
\[
g'(x) = e^x \cos x - e^x \sin x
\]
So, the derivative is:
\[
g'(x) = e^x (\cos x - \sin x)
\]

### Step 2: Solve \( g'(x) = 0 \)

To find the critical points, we solve:
\[
e^x (\cos x - \sin x) = 0
\]
Since \( e^x \) is never zero for any real \( x \), we need to solve:
\[
\cos x - \sin x = 0
\]
This simplifies to:
\[
\cos x = \sin x
\]
Dividing both sides by \( \cos x \) (assuming \( \cos x \neq 0 \)):
\[
1 = \tan x
\]
Thus, \( x = \frac{\pi}{4} + n\pi \) for some integer \( n \).

### Step 3: Find solutions in the interval \([0, 2\pi]\)

Now we find the values of \( x \) within the interval \( [0, 2\pi] \):
\[
x = \frac{\pi}{4}, \frac{5\pi}{4}
\]

### Step 4: Check the boundaries

Since the interval is closed, we also check the boundaries \( x = 0 \) and \( x = 2\pi \). However, \( g'(0) \) and \( g'(2\pi) \) do not result in zeros, so they are not critical points.

### Conclusion:

The critical points of \( g(x) = e^x \cos x \) on the interval \([0, 2\pi]\) are:
\[
x = \frac{\pi}{4}, \frac{5\pi}{4}
\]

Would you like further clarification or details on any step?

---

Here are 5 related questions for further exploration:
1. How would the critical points change if we considered the interval \([-2\pi, 2\pi]\)?
2. What is the behavior of the function \(g(x) = e^x \cos x\) near its critical points?
3. Can you find the inflection points of \(g(x)\) by taking the second derivative?
4. What happens to the function as \( x \to \infty \)?
5. How would you use the second derivative test to determine if the critical points are maxima or minima?

### Tip:
The product rule and chain rule are essential tools in differentiating composite functions, so make sure you are comfortable with both.

Find the critical points of g(x) = e^x cos x on the interval [0, 2π].

Explore how to find the critical points of the function g(x) = e^x cos x within the interval [0, 2π]. This problem involves using the product rule for derivatives and solving a trigonometric equation. Suitable for students studying Calculus 1.

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