Use the Second Derivative Test to Find all Relative Extrema f(x) = x^3  3x^2 + 2  Summary and Q&A
TL;DR
Use the second derivative test to find the relative maximums and minimums of a function.
Key Insights
 🏆 The second derivative test is used to find relative maximums and minimums of a function.
 😫 Critical numbers are found by setting the derivative of the function equal to zero.
 ❎ If the second derivative is negative at a critical number, it indicates a relative maximum.
 #️⃣ If the second derivative is positive at a critical number, it indicates a relative minimum.
 🏆 The second derivative test is based on the concavity of the function.
 🔌 Plugging the critical numbers back into the original function gives the actual maximum and minimum values.
 🏆 The second derivative test may be inconclusive if the second derivative is zero at a critical number.
Transcript
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Questions & Answers
Q: What is the first step in the second derivative test?
The first step is to find the critical numbers by setting the derivative of the function equal to zero.
Q: How do you determine if a critical number is a relative maximum or minimum?
You plug the critical number into the second derivative of the function. If the second derivative is negative, it is a relative maximum, and if it is positive, it is a relative minimum.
Q: What does it mean if the second derivative is zero at a critical number?
If the second derivative is zero at a critical number, the second derivative test is inconclusive, and you need to use other methods to determine if it is a maximum or minimum.
Q: Why is it important to plug the critical numbers back into the original function?
Plugging the critical numbers back into the original function gives the actual maximum or minimum values of the function.
Summary & Key Takeaways

The first step in the second derivative test is to find the critical numbers by setting the derivative of the function equal to zero.

Plug the critical numbers into the second derivative of the function.

If the second derivative is negative at a critical number, it indicates a relative maximum, and if it is positive, it indicates a relative minimum.