Step 3: Continuation Condition 4 - f'(n)=-n =ƒ'(6)=-6<0 The graph is decreasing at x=6 Condition 5 = ⇒ f'(n+4)=n+1 ⇒ƒ'(6+4)=6+1 ⇒ƒ(10)=7>0 The graph is increasing at x = 10 Condition 6 = f(n)=n -ƒ(6)=6 The graph passes through point (6,6) Condition 7 ⇒ƒ(6+4)=6+7 ⇒ƒ(10)=13 The graph passes through point (10,13) Step 4: Draw the graph of the function f(x) The graph of the function f(x) Concave down f" (x)<0 x 6 (6,6) (10, 13) concave up F(x)>0 Concave down x > 10 Step 1: Define the problem The given condtions are • f'(x) <0,x0,nn+4 • f'(n) = -n • f'(n+4)=n+1 • f(n)=n • f(n+4)=n+7 Step 2: Determine the all important points of function with given conditions Given that n = 6 Condition 1 ⇒f" (x) <0,x0,n0,6 10 The graph is concave down over the interval(10,∞) The concavity of graph changes at x=6,10 then these are inflection points of function f(x).
Step 3: Continuation Condition 4 - f'(n)=-n =ƒ'(6)=-6<0 The graph is decreasing at x=6 Condition 5 = ⇒ f'(n+4)=n+1 ⇒ƒ'(6+4)=6+1 ⇒ƒ(10)=7>0 The graph is increasing at x = 10 Condition 6 = f(n)=n -ƒ(6)=6 The graph passes through point (6,6) Condition 7 ⇒ƒ(6+4)=6+7 ⇒ƒ(10)=13 The graph passes through point (10,13) Step 4: Draw the graph of the function f(x) The graph of the function f(x) Concave down f" (x)<0 x 6 (6,6) (10, 13) concave up F(x)>0 Concave down x > 10 Step 1: Define the problem The given condtions are • f'(x) <0,x0,nn+4 • f'(n) = -n • f'(n+4)=n+1 • f(n)=n • f(n+4)=n+7 Step 2: Determine the all important points of function with given conditions Given that n = 6 Condition 1 ⇒f" (x) <0,x0,n0,6 10 The graph is concave down over the interval(10,∞) The concavity of graph changes at x=6,10 then these are inflection points of function f(x).
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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Question
Mathematically show to find the slope and how to check the listed inflection points. Additionally shows steps on how to check for the points and prove that they are true.
Transcribed Image Text:Step 3: Continuation
Condition 4
- f'(n)=-n
=ƒ'(6)=-6<0
The graph is decreasing at x=6
Condition 5
=
⇒ f'(n+4)=n+1
⇒ƒ'(6+4)=6+1
⇒ƒ(10)=7>0
The graph is increasing at x = 10
Condition 6
= f(n)=n
-ƒ(6)=6
The graph passes through point (6,6)
Condition 7
⇒ƒ(6+4)=6+7
⇒ƒ(10)=13
The graph passes through point (10,13)
Step 4: Draw the graph of the function f(x)
The graph of the function f(x)
Concave
down
f" (x)<0
x 6
(6,6)
(10, 13)
concave
up
F(x)>0
Concave
down
x > 10
Transcribed Image Text:Step 1: Define the problem
The given condtions are
• f'(x) <0,x<n
•
f(x)>0,n<x<n+4
f(x)<0,x>n+4
• f'(n) = -n
• f'(n+4)=n+1
• f(n)=n
• f(n+4)=n+7
Step 2: Determine the all important points of function with given conditions
Given that n = 6
Condition 1
⇒f" (x) <0,x<n
⇒ f'(x) <0,x<6
It means graph is concave down over the interval (-∞,6)
Condition 2
f(x)>0,n<x<n+4
⇒ f'(x)>0,6 <x<10
The graph is concave up over the interval (6,10)
Condition 3
⇒ f'(x) <0,x> 10
The graph is concave down over the interval(10,∞)
The concavity of graph changes at x=6,10 then these are inflection points of function f(x).
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