# Are the inflection points where f'(x) = zero or where the graph changes from concave up to concave down?

The inflection point is a point where the graph of the function changes from concave up to concave down or vice versa.

To find this we can graph the function:

graph{42x^5 [-3.894, 3.897, -1.95, 1.948]}

Note It is important to check to see whether concavity actually changes.

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I have been taught and, following our textbook's lead, I continue to teach , that an inflection point is a point on the graph at which the concavity changes.

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The inflection points of a function occur where the second derivative changes sign, indicating a change in concavity. They are not necessarily where the first derivative is zero, but rather where the graph transitions from concave up to concave down, or vice versa.

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Inflection points occur where the graph changes from concave up to concave down or vice versa. Therefore, they are not necessarily where the derivative, f'(x), equals zero.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you graph #f(x)=x^3-3x^2-9x+6# using the information given by the first derivative?
- What are the points of inflection, if any, of #f(x)=x^4-x^3+6 #?
- What is different between critical point and inflection point?
- How do you find the first and second derivative of #(ln(x^2-1))/x^2#?
- Is #f(x)=e^(3-3x)+x/ln2x# concave or convex at #x=1#?

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