Using Second Derivatives


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You can use the second derivative of a function to find sections of the curve that are convex or concave or find points of inflection. You can use the following facts to solve different problems:

Note: Before looking at this, refer to the guide for textbook reference 12.9 because this guide uses concepts from there.

Example 1:

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Answer 1
Example 2
Answer 2

Example 2:

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Answer 1
Example 2
Answer 2

For points of inflection, you form a equation using f''(x)=0 and solve for x. Do not forget to find the corresponding y coordinate or you will lose a mark.


Example 3:

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Example 2
Answer 2

For some cases you may get multiple values of x after solving the equation. This doesn't mean that all of them are points of inflection. To prove that they are points of inflection, you will have to look at points at either side like you did in AS. If there is a change in sign, then the points are points of inflection.

Practice Question:





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Note: When doing questions like part b) you need to chose values that don't cross over the x coordinates you are testing. For example for x=1/4, x=1/3 is used for the value greater than x=1/4 instead of a value like x=3/5 which is larger than the other x coordinate obtained after solving the equation, x=1/2. Similarly for x=1/4, x=2/5 is used for the value smaller than x=1/2 instead of a value like x=1/10 which is smaller than x=1/4.