A point on the curve `y= x^(3)-3x + 5` at which the tangent line is parallel to `y= -2x` is

Options

A. (1,3)
B. (0,5)
C. `((1)/(sqrt3), 5- (8 sqrt3)/(9))`
D. `((1)/(sqrt2), 0)`

Answer

To find the point on the curve \( y = x^3 - 3x + 5 \) where the tangent line is parallel to the line \( y = -2x \), we need to follow these steps:

### Step 1: Find the derivative of the curve
The derivative \( \frac{dy}{dx} \) gives us the slope of the tangent line at any point on the curve.

Given:
\[ y = x^3 - 3x + 5 \]

Differentiating with respect to \( x \):
\[ \frac{dy}{dx} = 3x^2 - 3 \]

### Step 2: Set the derivative equal to the slope of the given line
Since the tangent line is parallel to \( y = -2x \), the slope of the tangent line must be \( -2 \). Therefore, we set the derivative equal to \( -2 \):
\[ 3x^2 - 3 = -2 \]

### Step 3: Solve for \( x \)
Rearranging the equation:
\[ 3x^2 - 3 + 2 = 0 \]
\[ 3x^2 - 1 = 0 \]
\[ 3x^2 = 1 \]
\[ x^2 = \frac{1}{3} \]
\[ x = \pm \frac{1}{\sqrt{3}} \quad \text{(or } x = \pm \frac{\sqrt{3}}{3} \text{)} \]

### Step 4: Find the corresponding \( y \) values
Now we will find the \( y \) values for both \( x = \frac{1}{\sqrt{3}} \) and \( x = -\frac{1}{\sqrt{3}} \).

**For \( x = \frac{1}{\sqrt{3}} \)**:
\[ y = \left(\frac{1}{\sqrt{3}}\right)^3 - 3\left(\frac{1}{\sqrt{3}}\right) + 5 \]
\[ = \frac{1}{3\sqrt{3}} - \frac{3\sqrt{3}}{3} + 5 \]
\[ = \frac{1}{3\sqrt{3}} - \sqrt{3} + 5 \]
\[ = 5 - \sqrt{3} + \frac{1}{3\sqrt{3}} \]
To simplify, we can write:
\[ = 5 - \frac{3\sqrt{3}}{3} + \frac{1}{3\sqrt{3}} = 5 - \frac{8\sqrt{3}}{3} \]

**For \( x = -\frac{1}{\sqrt{3}} \)**:
\[ y = \left(-\frac{1}{\sqrt{3}}\right)^3 - 3\left(-\frac{1}{\sqrt{3}}\right) + 5 \]
\[ = -\frac{1}{3\sqrt{3}} + \sqrt{3} + 5 \]
\[ = 5 + \sqrt{3} - \frac{1}{3\sqrt{3}} \]
\[ = 5 + \frac{3\sqrt{3}}{3} - \frac{1}{3\sqrt{3}} = 5 + \frac{8\sqrt{3}}{3} \]

### Step 5: Write the points
Thus, the points on the curve where the tangent line is parallel to \( y = -2x \) are:
1. \( \left(\frac{1}{\sqrt{3}}, 5 - \frac{8\sqrt{3}}{3}\right) \)
2. \( \left(-\frac{1}{\sqrt{3}}, 5 + \frac{8\sqrt{3}}{3}\right) \)