Find the critical numbers and then the critical points of the functions. 3x². = - 2x +2 3 +1 Skech the graph of the following functions, by finding and the intervals of the local mins and maximums increase and decrease and making a sign diagram for the first derivative. 3 2 x²³² + 3x²² - 92-3
Find the critical numbers and then the critical points of the functions. 3x². = - 2x +2 3 +1 Skech the graph of the following functions, by finding and the intervals of the local mins and maximums increase and decrease and making a sign diagram for the first derivative. 3 2 x²³² + 3x²² - 92-3
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter1: Expressions And Functions
Section: Chapter Questions
Problem 74SGR
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![**Critical Points and Graphing Functions**
**Objective:**
1. Find the critical numbers and then the critical points of the functions.
\[1. \quad y = 3x^2 - 2x^3 \]
\[2. \quad f(x) = (x^2/3) + 1 \]
2. Sketch the graph of the following functions by finding the local mins and maxs and the intervals of increase and decrease and making a sign diagram for the first derivative.
\[3. \quad y = 2x^3 - 3x^2 - 12x + 3 \]
**Instructions:**
1. For each function, determine the first derivative.
2. Identify critical points by setting the first derivative equal to zero and solving for \( x \).
3. Analyze the second derivative to determine concavity at the critical points.
4. Create a sign diagram to showcase intervals of increase and decrease.
5. Use the critical points and the sign diagram to sketch the graph of each function accurately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbd18e3e2-85c9-4b2f-a0fd-7822a7147daa%2Fd6539375-a753-4ca3-9a84-b6c2336d0a8d%2Fwxanf9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Critical Points and Graphing Functions**
**Objective:**
1. Find the critical numbers and then the critical points of the functions.
\[1. \quad y = 3x^2 - 2x^3 \]
\[2. \quad f(x) = (x^2/3) + 1 \]
2. Sketch the graph of the following functions by finding the local mins and maxs and the intervals of increase and decrease and making a sign diagram for the first derivative.
\[3. \quad y = 2x^3 - 3x^2 - 12x + 3 \]
**Instructions:**
1. For each function, determine the first derivative.
2. Identify critical points by setting the first derivative equal to zero and solving for \( x \).
3. Analyze the second derivative to determine concavity at the critical points.
4. Create a sign diagram to showcase intervals of increase and decrease.
5. Use the critical points and the sign diagram to sketch the graph of each function accurately.
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