### Horizontal Tangents of a Polynomial Function#### Problem DescriptionThe graph of the function \( f(x) = 2x^3 + 12x^2 - 72x + 14 \) has two horizontal tangents. One occurs at a negative value of \( x \) and the other at a positive value of \( x \).- **Question 1:** What is the negative value of \( x \) where a horizontal tangent occurs? *(Input box for the answer)*- **Question 2:** What is the positive value of \( x \) where a horizontal tangent occurs? *(Input box for the answer)*#### ExplanationTo find where the horizontal tangents occur, we need to find the critical points of the function \( f(x) = 2x^3 + 12x^2 - 72x + 14 \). Critical points occur where the derivative of the function equals zero. So, we calculate the first derivative \( f'(x) \) and solve for \( x \):1. Find the first derivative \( f'(x) \): \[ f'(x) = \frac{d}{dx} (2x^3 + 12x^2 - 72x + 14) = 6x^2 + 24x - 72 \]2. Set the derivative equal to zero and solve for \( x \): \[ 6x^2 + 24x - 72 = 0 \] Divide through by 6: \[ x^2 + 4x - 12 = 0 \]3. Factor the quadratic equation: \[ (x + 6)(x - 2) = 0 \]4. Solve for \( x \): \[ x = -6 \quad \text{or} \quad x = 2 \]Thus, the negative value of \( x \) where a horizontal tangent occurs is \(-6\), and the positive value is \(2\).

Name: ### Horizontal Tangents of a Polynomial Function#### Problem Descript
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: ### Horizontal Tangents of a Polynomial Function#### Problem DescriptionThe graph of the function \( f(x) = 2x^3 + 12x^2 - 72x + 14 \) has two horizontal tangents. One occurs at a negative value of \( x \) and the other at a positive value of \( x \