Find a point on the graph of `y=x^(3)`, where the tangent is parallel to the chord joining (1, 1) and (3, 27).

Answer

To find a point on the graph of \( y = x^3 \) where the tangent is parallel to the chord joining the points \( (1, 1) \) and \( (3, 27) \), we can follow these steps:

### Step 1: Find the slope of the chord
The slope of the chord joining the points \( (1, 1) \) and \( (3, 27) \) can be calculated using the formula:

\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \( (x_1, y_1) = (1, 1) \) and \( (x_2, y_2) = (3, 27) \).

Calculating the slope:

\[ \text{slope} = \frac{27 - 1}{3 - 1} = \frac{26}{2} = 13 \]

### Step 2: Find the derivative of the function
The derivative of the function \( y = x^3 \) gives us the slope of the tangent at any point on the curve.

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

### Step 3: Set the derivative equal to the slope of the chord
To find the points where the tangent is parallel to the chord, we set the derivative equal to the slope of the chord:

\[ 3x^2 = 13 \]

### Step 4: Solve for \( x \)
Now, we solve for \( x \):

\[ x^2 = \frac{13}{3} \]

Taking the square root of both sides:

\[ x = \pm \sqrt{\frac{13}{3}} \]

### Step 5: Find the corresponding \( y \) values
Now we need to find the corresponding \( y \) values for both \( x \) values using the original function \( y = x^3 \).

1. For \( x = \sqrt{\frac{13}{3}} \):

\[ y = \left(\sqrt{\frac{13}{3}}\right)^3 = \frac{13\sqrt{\frac{13}{3}}}{3} \]

2. For \( x = -\sqrt{\frac{13}{3}} \):

\[ y = \left(-\sqrt{\frac{13}{3}}\right)^3 = -\frac{13\sqrt{\frac{13}{3}}}{3} \]

### Step 6: Write the points
Thus, the points on the graph where the tangent is parallel to the chord are:

\[ \left(\sqrt{\frac{13}{3}}, \frac{13\sqrt{\frac{13}{3}}}{3}\right) \quad \text{and} \quad \left(-\sqrt{\frac{13}{3}}, -\frac{13\sqrt{\frac{13}{3}}}{3}\right) \]

### Summary of the solution
The points on the graph of \( y = x^3 \) where the tangent is parallel to the chord joining \( (1, 1) \) and \( (3, 27) \) are:

1. \( \left(\sqrt{\frac{13}{3}}, \frac{13\sqrt{\frac{13}{3}}}{3}\right) \)
2. \( \left(-\sqrt{\frac{13}{3}}, -\frac{13\sqrt{\frac{13}{3}}}{3}\right) \)