Find the intervals on which the graph of f is concave​ upward, the intervals on which the graph of f is concave​ downward, and the inflection points. ​f(x)=−x6+30x5−30x+3

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Find the intervals on which the graph of f is concave​ upward, the intervals on which the graph of f is concave​ downward, and the inflection points.
 
​f(x)=−x6+30x5−30x+3
 
 
 

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Part 1
For what​ interval(s) of x is the graph of f concave​ upward? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
 
 
A.
enter your response here
​(Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as​ needed.)
 
B.
The graph is never concave upward.
 
Part 2
For what​ interval(s) of x is the graph of f concave​ downward? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
 
 
A.
enter your response here
​(Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as​ needed.)
 
B.
The graph is never concave downward.
 
Part 3
Determine the x coordinates of any inflection points of the graph of​ f(x). Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
 
 
A.
x=enter your response here
​(Type an exact answer. Use a comma to separate answers as​ needed.)
 
B.
There are no inflection points.
 
 

 

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