Relative Extrema vs Absolute Extrema

In summary, the main difference between relative extrema and absolute extrema is that relative extrema are the greatest or smallest in their local neighborhood, while absolute extrema are the greatest or smallest anywhere in the domain. This means that absolute extrema are more involved to find, as they require checking critical points, poles, and boundaries of the function's domain.
  • #1
harpazo
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In basic terms, what is the main difference between relative extrema and absolute extrema? I know that absolute extrema is more involved but why is this the case?
 
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  • #2
A relative maximum (minimum) is the greatest (smallest) in its neighborhood but an absolute maximum (minimum) is the greatest (smallest) anywhere (in the domain). For example:

View attachment 6470

In the picture we see that the function has $3$ relative minima and an absolute minimum, which is smallest of all relative minima. It also has $2$ relative maxima, but it has no absolute maximum, since at the boundaries the function goes to $+\infty$.
In general, to find an absolute extrema, besides the critical points we have to check also the poles of the function and the boundaries of its domain.
 

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  • #3
mathmari said:
A relative maximum (minimum) is the greatest (smallest) in its neighborhood but an absolute maximum (minimum) is the greatest (smallest) anywhere (in the domain). For example:



In the picture we see that the function has $3$ relative minima and an absolute minimum, which is smallest of all relative minima. It also has $2$ relative maxima, but it has no absolute maximum, since at the boundaries the function goes to $+\infty$.
In general, to find an absolute extrema, besides the critical points we have to check also the poles of the function and the boundaries of its domain.

Very cool. Nice picture.
 

FAQ: Relative Extrema vs Absolute Extrema

What is the difference between relative extrema and absolute extrema?

Relative extrema refers to the highest or lowest points on a graph within a specific interval, while absolute extrema refers to the overall highest or lowest point of a function.

How do you find relative extrema?

To find relative extrema, you must first take the derivative of the function and set it equal to zero. Then, solve for the x-values where the derivative is equal to zero. These x-values will be the coordinates of the relative extrema on the original function.

What is the significance of relative extrema?

Relative extrema can help identify where a function changes from increasing to decreasing or vice versa. They can also help determine the overall shape and behavior of a function.

Can a function have multiple relative extrema?

Yes, a function can have multiple relative extrema. These points can be local maximums or minimums, depending on the behavior of the function around those points.

How do you differentiate between a relative and absolute maximum or minimum?

A relative maximum or minimum is a point where the function changes from increasing to decreasing or vice versa, while an absolute maximum or minimum is the overall highest or lowest point of the function. Relative extrema can be found by taking the derivative of the function, while absolute extrema can be found by analyzing the endpoints of the function's domain or using the First Derivative Test.

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