The function given is \( y = 3x^5 - 5x^3 \).

Let's analyze this function.

### 1. **Finding the first derivative:**
The first derivative of the function \( y \) with respect to \( x \) will give us the slope of the tangent to the curve at any point \( x \).

\[
\frac{dy}{dx} = 15x^4 - 15x^2
\]

### 2. **Finding the critical points:**
Critical points occur where the first derivative is zero or undefined. Setting the first derivative equal to zero:

\[
15x^4 - 15x^2 = 0
\]

Factor the equation:

\[
15x^2(x^2 - 1) = 0
\]

Further factoring gives:

\[
15x^2(x - 1)(x + 1) = 0
\]

So, the critical points are \( x = 0 \), \( x = 1 \), and \( x = -1 \).

### 3. **Second derivative to determine concavity:**
The second derivative will help us determine the concavity of the function and whether the critical points are maxima, minima, or points of inflection.

\[
\frac{d^2y}{dx^2} = 60x^3 - 30x
\]

Evaluate the second derivative at the critical points:

- At \( x = 0 \):

\[
\frac{d^2y}{dx^2} = 60(0)^3 - 30(0) = 0
\]

Since the second derivative is zero, \( x = 0 \) might be a point of inflection.

- At \( x = 1 \):

\[
\frac{d^2y}{dx^2} = 60(1)^3 - 30(1) = 60 - 30 = 30 \quad (\text{concave up, indicating a local minimum})
\]

- At \( x = -1 \):

\[
\frac{d^2y}{dx^2} = 60(-1)^3 - 30(-1) = -60 + 30 = -30 \quad (\text{concave down, indicating a local maximum})
\]

### 4. **Finding the points:**
Now, let's find the corresponding \( y \) values for the critical points:

- At \( x = 0 \):

\[
y = 3(0)^5 - 5(0)^3 = 0
\]

So, \( (0, 0) \) is potentially an inflection point.

- At \( x = 1 \):

\[
y = 3(1)^5 - 5(1)^3 = 3 - 5 = -2
\]

So, \( (1, -2) \) is a local minimum.

- At \( x = -1 \):

\[
y = 3(-1)^5 - 5(-1)^3 = -3 + 5 = 2
\]

So, \( (-1, 2) \) is a local maximum.

### 5. **Summary:**
- The function has a local maximum at \( (-1, 2) \).
- The function has a local minimum at \( (1, -2) \).
- The point \( (0, 0) \) could be an inflection point.

Would you like more details on any of these steps or have any questions?

Here are 5 related questions:
1. How do you confirm whether \( x = 0 \) is an inflection point?
2. What are the intervals of concavity for the function \( y = 3x^5 - 5x^3 \)?
3. Can you graph the function \( y = 3x^5 - 5x^3 \) and indicate the local maxima, minima, and inflection points?
4. How does the function \( y = 3x^5 - 5x^3 \) behave as \( x \) approaches positive and negative infinity?
5. What is the significance of inflection points in the graph of a function?

**Tip:** When analyzing higher-degree polynomial functions, always check for potential inflection points where the second derivative is zero, as these can indicate changes in concavity.

Explore the function y = 3x^5 - 5x^3 with detailed derivative analysis. Learn how to find critical points, determine concavity, and identify inflection points. Suitable for advanced high school and college students.

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