1. Find the line equation D that touches the graph D: y = x³ - 4x at point A (1, 3), followed by y = f '(x0) (x- x0) + y0.   Find the coordinates of another intersection between line D and graph (C).
Please tell me

To solve this problem, we need to follow a few steps. Let's go through them one by one.

### Step 1: Find the Derivative of the Function
The function given is \( y = x^3 - 4x \). To find the slope of the tangent line at the point \( A(1, 3) \), we need to compute the derivative of \( y \) with respect to \( x \).

\[
y' = \frac{dy}{dx} = 3x^2 - 4
\]

### Step 2: Find the Slope at the Point \( A(1, 3) \)
Now, substitute \( x = 1 \) into the derivative to find the slope of the tangent line at \( x = 1 \).

\[
y'(1) = 3(1)^2 - 4 = 3 - 4 = -1
\]

So, the slope of the tangent line at the point \( A(1, 3) \) is \(-1\).

### Step 3: Find the Equation of the Tangent Line
The general equation for the tangent line is given by:

\[
y = f'(x_0) (x - x_0) + y_0
\]

Here, \( x_0 = 1 \), \( y_0 = 3 \), and \( f'(x_0) = -1 \). Substituting these values into the equation:

\[
y = -1(x - 1) + 3
\]

Simplifying this:

\[
y = -x + 1 + 3 = -x + 4
\]

So, the equation of the tangent line \( D \) is:

\[
y = -x + 4
\]

### Step 4: Find the Other Intersection Point Between the Line \( D \) and the Curve
To find the other intersection point, we need to solve the system of equations formed by the line \( y = -x + 4 \) and the curve \( y = x^3 - 4x \).

Set the equations equal to each other:

\[
x^3 - 4x = -x + 4
\]

This simplifies to:

\[
x^3 - 3x - 4 = 0
\]

### Step 5: Solve the Cubic Equation
The equation \( x^3 - 3x - 4 = 0 \) is a cubic equation. We already know one root, \( x = 1 \), since the tangent touches the curve at this point. We can use synthetic division or factoring to find the other roots.

Let's factorize the cubic equation:

\[
x^3 - 3x - 4 = (x - 1)(x^2 + x + 4)
\]

Now, solve the quadratic equation \( x^2 + x + 4 = 0 \):

\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(4)}}{2(1)} = \frac{-1 \pm \sqrt{1 - 16}}{2} = \frac{-1 \pm \sqrt{-15}}{2}
\]

The solutions are complex numbers, meaning there are no other real intersection points. Therefore, the only real intersection between the tangent line \( D \) and the curve \( y = x^3 - 4x \) is at the point \( A(1, 3) \).

Would you like further details or have any other questions?

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Here are 5 related questions to consider:

1. How do we solve cubic equations with real roots?
2. Can there be more than one tangent line at a given point on a curve?
3. What is the geometric significance of a tangent line to a curve?
4. How do complex roots affect the intersection points between a line and a curve?
5. What is the process of finding the tangent line to a curve at any given point?

**Tip:** For cubic equations, understanding how to factorize or apply synthetic division is crucial in finding real roots efficiently.

Explore how to find the tangent line equation to the curve y = x^3 - 4x at point (1, 3), and discover the intersection point with detailed steps. This problem involves calculus, derivatives, and cubic equations. Suitable for undergraduate-level students and learners interested in calculus concepts.

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