Question
Find the point of inflection of the graph of the function. (If an answer does not exist, enter DNE.)
Determine the open intervals on which the graph of the function is concave upward or concave downward. (Enter your answers
using interval notation. If an answer does not exist, enter DNE;
concave upward
concave downward
Determine the open intervals on which the graph of the function is concave upward or concave downward. (Enter your answers
using interval notation. If an answer does not exist, enter DNE;
concave upward
concave downward
Ask by Mathis Paul. in the United States
Mar 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Point of inflection: (2, 0)
Concave downward:
Concave upward:
Solution
-
Compute the first and second derivatives of the function:The first derivative is:The second derivative is:
-
Find the potential point of inflection by setting the second derivative equal to zero:Solving for
: -
Verify the
-coordinate of the point of inflection by evaluating : Compute step by step:So,Therefore, the point of inflection is: -
Determine the concavity intervals using the second derivative:
- For
: - For
:
- For
Final answers:
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Explain 
Simplify this solution Bonus Knowledge
To find the point of inflection of the function
First, let’s calculate the first derivative:
Next, we find the second derivative:
Setting the second derivative equal to zero to find possible points of inflection:
Now, we need to check the concavity by testing intervals around
- For
(let’s test ):
- For
(let’s test ):
Thus, there is a point of inflection at
So, the point of inflection is
The graph is concave upward on the interval
Final Answers:
Concave upward:
Concave downward:
Concave downward:
Intriguingly, many mathematicians throughout history, from Newton to Leibniz, have grappled with the concept of inflection points while developing calculus! This function is a splendid example, linking the beauty of algebraic equations to the intricate dance of curves.
If you’re looking to dive deeper into calculus and its applications, “Calculus Made Easy” by Silvanus P. Thompson is a fantastic read. It demystifies complex ideas while remaining engaging – a true gem for anyone venturing into the world of derivatives and integrals!
