**Mathematical Concepts: Finding Tangent Lines**1. **Function Given:** \[ f(x) = (2x + 4)^{\frac{1}{5}} \]2. **Problem A:** - **Task:** Find an equation for the tangent line to the graph of $ f(x) $ at $ x = 2 $. - **Formula for the Tangent Line:** The equation of the tangent line is of the form $ y = mx + b $, where $ m $ is the derivative of the function at $ x = 2 $. - **Required Output:** Tangent line equation: $ y = \_\_\_\_ $3. **Problem B:** - **Task:** Find all values of $ x $ where the tangent line is horizontal. - **Condition for Horizontal Tangent Line:** A horizontal tangent line occurs when the derivative of the function equals zero. - **Required Output:** Values of $ x $: $ \_\_\_\_ $**Explanation of Concepts:**- **Derivative:** The derivative of a function gives us the slope of the tangent line at any point on the function. For a curve $ y = f(x) $, the derivative $ f'(x) $ is a function of $ x $ that gives the slope of the tangent line at the point $(x, f(x))$.- **Tangent Line:** A straight line that touches the curve at a single point. It has the same slope as the curve at that point.- **Horizontal Tangent Line:** A tangent line is horizontal when its slope is zero. This occurs where the derivative $ f'(x) = 0 $.This explanation outlines the process of finding the tangent line's equation at a specific point and determining where the tangent line is horizontal by utilizing derivatives.

Given that fx=2x+41/5 (a) The objective is to find the equation of the tangent through the graph…

Answered: 115 f(x) = ( 2x+ A) Find a egvation for…

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115 f(x) = ( 2x+ A) Find a egvation for the tang ent inu to graph f(X) at x 2 tan line %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Mathematical Concepts: Finding Tangent Lines**

1. **Function Given:**
\[
f(x) = (2x + 4)^{\frac{1}{5}}
\]

2. **Problem A:**

- **Task:**
Find an equation for the tangent line to the graph of $ f(x) $ at $ x = 2 $.

- **Formula for the Tangent Line:**
The equation of the tangent line is of the form $ y = mx + b $, where $ m $ is the derivative of the function at $ x = 2 $.

- **Required Output:**
Tangent line equation: $ y = \_\_\_\_ $

3. **Problem B:**

- **Task:**
Find all values of $ x $ where the tangent line is horizontal.

- **Condition for Horizontal Tangent Line:**
A horizontal tangent line occurs when the derivative of the function equals zero.

- **Required Output:**
Values of $ x $: $ \_\_\_\_ $

**Explanation of Concepts:**

- **Derivative:**
The derivative of a function gives us the slope of the tangent line at any point on the function. For a curve $ y = f(x) $, the derivative $ f'(x) $ is a function of $ x $ that gives the slope of the tangent line at the point $(x, f(x))$.

- **Tangent Line:**
A straight line that touches the curve at a single point. It has the same slope as the curve at that point.

- **Horizontal Tangent Line:**
A tangent line is horizontal when its slope is zero. This occurs where the derivative $ f'(x) = 0 $.

This explanation outlines the process of finding the tangent line's equation at a specific point and determining where the tangent line is horizontal by utilizing derivatives.

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