**Problem 5: Finding Critical Numbers Using Calculus**Let $ f(x) = x - 3x^{2/3} $. Use calculus to determine the critical numbers for the function.**Instructions for Solving:**To find the critical numbers, follow these steps:1. **Differentiate the Function:** - Find the derivative $ f'(x) $ of the function $ f(x) = x - 3x^{2/3} $.2. **Set the Derivative to Zero:** - Solve for $ x $ when $ f'(x) = 0 $.3. **Identify Undefined Points:** - Determine any points where $ f'(x) $ is undefined.4. **Solution:** - Critical numbers are the values of $ x $ found in steps 2 and 3.These steps will lead you to identify all points $ x $ where the slope of the tangent is zero or undefined, indicating potential maxima, minima, or inflection points.

Consider the provided function, fx=x-3x23 Find the critical point of the critical point. Find the…

Answered: 5. Let f(x) = x-3x73. Use calculus to…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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**Problem 5: Finding Critical Numbers Using Calculus**

Let $ f(x) = x - 3x^{2/3} $. Use calculus to determine the critical numbers for the function.

**Instructions for Solving:**

To find the critical numbers, follow these steps:

1. **Differentiate the Function:**
- Find the derivative $ f'(x) $ of the function $ f(x) = x - 3x^{2/3} $.

2. **Set the Derivative to Zero:**
- Solve for $ x $ when $ f'(x) = 0 $.

3. **Identify Undefined Points:**
- Determine any points where $ f'(x) $ is undefined.

4. **Solution:**
- Critical numbers are the values of $ x $ found in steps 2 and 3.

These steps will lead you to identify all points $ x $ where the slope of the tangent is zero or undefined, indicating potential maxima, minima, or inflection points.

Expert Solution

Step 1

Consider the provided function,

$f (x) = x - 3 x^{\frac{2}{3}}$

Find the critical point of the critical point.

Find the derivative of the function.

$\begin{array}{rcl} f' (x) & = & \frac{d}{d x} [x - 3 x^{\frac{2}{3}}] \\ = & 1 - 3 \cdot \frac{2}{3} x^{\frac{2}{3} - 1} \\ = & 1 - 2 x^{- \frac{1}{3}} \end{array}$

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