To analyze the behavior of the function \( y = x - x^3 \) at the point \( (1, 0) \), we'll explore several key aspects:

### 1. **Check if (1, 0) lies on the curve:**
We substitute \( x = 1 \) into the equation to verify if \( y = 0 \):

\[
y = 1 - 1^3 = 1 - 1 = 0
\]

So, \( (1, 0) \) lies on the curve.

### 2. **Find the derivative (slope of the tangent line):**
The derivative of \( y = x - x^3 \) will give us the slope of the tangent line at \( x = 1 \).

\[
\frac{dy}{dx} = 1 - 3x^2
\]

Now, evaluate the derivative at \( x = 1 \):

\[
\frac{dy}{dx}\bigg|_{x=1} = 1 - 3(1)^2 = 1 - 3 = -2
\]

So, the slope of the tangent line at \( (1, 0) \) is \( -2 \).

### 3. **Equation of the tangent line:**
Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) = (1, 0) \):

\[
y - 0 = -2(x - 1)
\]

Simplifying:

\[
y = -2x + 2
\]

Thus, the equation of the tangent line at \( (1, 0) \) is \( y = -2x + 2 \).

### 4. **Concavity at \( x = 1 \) (second derivative):**
The second derivative will help us understand the concavity of the function at this point:

\[
\frac{d^2y}{dx^2} = -6x
\]

Evaluate the second derivative at \( x = 1 \):

\[
\frac{d^2y}{dx^2}\bigg|_{x=1} = -6(1) = -6
\]

Since the second derivative is negative, the function is concave down at \( x = 1 \), meaning \( (1, 0) \) is a local maximum.

### Summary:
- The point \( (1, 0) \) lies on the curve.
- The slope of the tangent line at \( (1, 0) \) is \( -2 \).
- The equation of the tangent line is \( y = -2x + 2 \).
- The function is concave down at \( (1, 0) \), indicating a local maximum.

Would you like further details on any of these points, or do you have any specific questions?

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Here are 5 related questions:
1. How would the tangent line change if evaluated at different points on the curve?
2. Can we determine other critical points of \( y = x - x^3 \)?
3. What is the significance of concavity in determining the behavior of a function?
4. How do you find inflection points of this function?
5. How can we approximate the function \( y = x - x^3 \) using a Taylor series?

**Tip:** Knowing how to differentiate functions helps in finding not only slopes but also understanding the curvature and critical points of graphs.

Explore the behavior of the function y = x - x^3 at the point (1, 0), including the tangent line equation, slope, and concavity. This detailed calculus problem covers derivatives, concavity, and local maxima, suitable for advanced high school students (Grades 11-12).

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