Introduction to minimum and maximum points | Functions | Algebra I | Khan Academy

TL;DR

Maximum and minimum points on a graph can be absolute or relative. Absolute points occur at the endpoints of an interval, while relative points are local extrema - higher or lower than their surrounding values.

Transcript

So right over here I've graphed the function y is equal to f of x. I've graphed over this interval. It looks like it's between 0 and some positive value. And I want to think about the maximum and minimum points on this. So we've already talked a little bit about absolute maximum and absolute minimum points on an interval. And those are pretty obvio... Read More

Key Insights

• 😥 Maximum and minimum points on a graph can be classified as either absolute or relative.
• 😥 Absolute points occur at the endpoints of an interval, while relative points are local extrema determined by the values of the surrounding points.
• 😥 Relative maximum points are higher than their neighboring points but not necessarily the highest in the interval.
• 😘 Relative minimum points are lower than their neighboring points but not necessarily the lowest in the interval.
• 😥 The definitions of relative extrema provide a more formal and rigorous way to determine these points mathematically.
• 👈 For a point to be a relative maximum, the function at that point should be greater than or equal to the values of x within an open interval around it.

Questions & Answers

Q: What defines an absolute maximum point on a graph in a given interval?

An absolute maximum point occurs at the beginning of an interval when it reaches the highest value of the function, denoted as f(a) or f(0) in this case.

Q: How is a relative maximum point different from an absolute maximum point?

A relative maximum point, like point c, occurs when the function takes on a higher value than the surrounding points but is not the highest value in the entire interval.

Q: Can there be multiple open intervals where a relative maximum point exists?

Yes, there can be multiple open intervals where a relative maximum point can be found. The definition states that for all x values within an open interval of c-h to c+h, the function at c is greater than or equal to any other value.

Q: How is a relative minimum point determined on a graph?

A relative minimum point, such as point d, is identified when the function reaches a lower value compared to the surrounding points but is not the lowest value in the entire interval.

Summary & Key Takeaways

• Absolute maximum points occur at the beginning of the interval, while absolute minimum points occur at the end.

• Relative maximum points, such as point c, are not the largest values but higher than surrounding points.

• Relative minimum points, like point d, are lower than surrounding points but not the lowest in the entire interval.