Calculate the coordinaties of the poinc
for which the congent 0 f
is perpendicular to the ling

Ask by Osborne Hardy. in South Africa

Jan 23,2025

Question

Calculate the coordinaties of the poinc for which the congent 0 f $$ f(x)=x^{3}-3 x+5 x+4 $$ is perpendicular to the ling $$ y=-\frac{1}{14} x+1 $$ $$ f^{\prime}(x)=3 x^{2}-6 x+5 $$

UpStudy AI · Accepted Answer

To find the coordinates of the points where the tangent line to the function $$ f(x) = x^{3} - 3x^{2} + 5x + 4 $$ is perpendicular to the line $$ y = -\frac{1}{14}x + 1 $$, follow these steps:

### 1. **Understand Perpendicular Slopes**

Two lines are perpendicular if the product of their slopes is $$-1$$.

Given the line:
$$ y = -\frac{1}{14}x + 1 $$
- **Slope ($$ m_1 $$)**: $$-\frac{1}{14}$$

Let the slope of the tangent line to $$ f(x) $$ be $$ m_2 = f'(x) $$. For the lines to be perpendicular:
$$ m_1 \times m_2 = -1 $$
$$ -\frac{1}{14} \times f'(x) = -1 $$
$$ f'(x) = 14 $$

### 2. **Find the Derivative of $$ f(x) $$**

Given:
$$ f(x) = x^{3} - 3x^{2} + 5x + 4 $$

Differentiate $$ f(x) $$:
$$ f'(x) = 3x^{2} - 6x + 5 $$

### 3. **Set the Derivative Equal to 14 and Solve for $$ x $$**

$$ 3x^{2} - 6x + 5 = 14 $$
$$ 3x^{2} - 6x - 9 = 0 $$
$$ x^{2} - 2x - 3 = 0 $$ *(Divided the entire equation by 3 for simplicity)*

Solve the quadratic equation:
$$ x = \frac{2 \pm \sqrt{4 + 12}}{2} $$
$$ x = \frac{2 \pm \sqrt{16}}{2} $$
$$ x = \frac{2 \pm 4}{2} $$

Thus:
$$ x = 3 \quad \text{or} \quad x = -1 $$

### 4. **Find the Corresponding $$ y $$-Coordinates**

Plug these $$ x $$-values back into $$ f(x) $$:

- **For $$ x = 3 $$**:
  $$ f(3) = 3^{3} - 3(3)^{2} + 5(3) + 4 = 27 - 27 + 15 + 4 = 19 $$
  $$ \text{Point: } (3, 19) $$

- **For $$ x = -1 $$**:
  $$ f(-1) = (-1)^{3} - 3(-1)^{2} + 5(-1) + 4 = -1 - 3 - 5 + 4 = -5 $$
  $$ \text{Point: } (-1, -5) $$

### 5. **Conclusion**

The coordinates of the points where the tangent to $$ f(x) $$ is perpendicular to the given line are:

- $$ (3, \, 19) $$
- $$ (-1, \, -5) $$
The tangent lines to $$ f(x) = x^{3} - 3x^{2} + 5x + 4 $$ that are perpendicular to the line $$ y = -\frac{1}{14}x + 1 $$ occur at the points $$ (3, 19) $$ and $$ (-1, -5) $$.

Calculate the coordinaties of the poinc
for which the congent 0 f
is perpendicular to the ling

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Calculate the coordinaties of the poinc for which the congent 0 f is perpendicular to the ling