What is different between critical point and inflection point?

Answer 1

There seem to be two definitions of "critical point" in use. But with that in mind:

A critical number for function #f#, is a number in the domain of #f# where #f'(x) = 0# or #f'(x)# does not exist. These are the numbers at which Fermat's Theorem on local extrema tells us #f# could have a local maximum or local minimum value.
In some usages "critical point" is synonymous with this "critical number". In other usages a critical point is a point #(x, f(x))# with #f'(x)=0# or #f'(x)# does not exist.
An inflection point for the graph of function #f# is a point on the graph at which the concavity of the graph changes. For twice differentiable functions, this is a point on the graph of #f# at which #f''(x)# changes sign.
Students sometimes use "inflection point" to mean an #x# value at which the concavity changes.
The difference is illustrated by #f(x) = 1/x# which is concave down on #(-oo, 0)# and concave up on #(0, oo)#. In the "point" usage, the graph has no inflection point, because there is no point on the graph where concavity changes.
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Answer 2

A critical point is a point on the graph of a function where its derivative is either zero or undefined. It indicates a potential location for a maximum, minimum, or saddle point.

An inflection point, on the other hand, is a point on the graph of a function where the concavity changes. It is where the second derivative of the function changes sign, indicating a change in the direction of curvature of the graph.

In summary, a critical point relates to the behavior of the first derivative of a function, indicating possible extrema, while an inflection point relates to the behavior of the second derivative, indicating where the curvature of the function changes.

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Answer from HIX Tutor

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.

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.

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