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 will 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:

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

Substituting the coordinates:

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

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

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

### Step 3: Set the derivative equal to the slope of the chord
To find the point 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:

\[ x = \pm \sqrt{\frac{13}{3}} = \pm \frac{\sqrt{39}}{3} \]

### Step 5: Find the corresponding \( y \) values
Now we substitute \( x \) back into the original equation \( y = x^3 \) to find the corresponding \( y \) values.

For \( x = \frac{\sqrt{39}}{3} \):

\[ y = \left(\frac{\sqrt{39}}{3}\right)^3 = \frac{39\sqrt{39}}{27} = \frac{13\sqrt{39}}{9} \]

For \( x = -\frac{\sqrt{39}}{3} \):

\[ y = \left(-\frac{\sqrt{39}}{3}\right)^3 = -\frac{39\sqrt{39}}{27} = -\frac{13\sqrt{39}}{9} \]

### Step 6: Write the points
Thus, the points on the graph of \( y = x^3 \) where the tangent is parallel to the chord are:

\[ \left(\frac{\sqrt{39}}{3}, \frac{13\sqrt{39}}{9}\right) \quad \text{and} \quad \left(-\frac{\sqrt{39}}{3}, -\frac{13\sqrt{39}}{9}\right) \]

### Summary
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. \( \left(\frac{\sqrt{39}}{3}, \frac{13\sqrt{39}}{9}\right) \)
2. \( \left(-\frac{\sqrt{39}}{3}, -\frac{13\sqrt{39}}{9}\right) \)

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