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What is the Local Maximum of a Function?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
A local maximum of a function is a point where the function's value is greater than or equal to the values at all nearby points. Imagine a small 'peak' or 'hilltop' on a graph; that peak is a local maximum. It's the highest point within a specific small region, but not necessarily the highest point overall.
Simple Example
Quick Example
Think about the number of runs scored by a cricket team in each over of a T20 match. If in the 10th over, the team scored 15 runs, and in the 9th and 11th overs they scored 8 and 12 runs respectively, then 15 runs in the 10th over is a local maximum. It was the highest score in that immediate part of the game, even if they scored 20 runs in the 18th over.
Worked Example
Step-by-Step
Let's find local maximums for the following set of daily temperatures (in Celsius) recorded over a week: Day 1: 28, Day 2: 30, Day 3: 29, Day 4: 32, Day 5: 31, Day 6: 34, Day 7: 33.
Step 1: Look at Day 1 (28). Is it greater than its neighbours? Only Day 2 (30) is its neighbour. 28 is not > 30. So, Day 1 is not a local maximum.
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Step 2: Look at Day 2 (30). Its neighbours are Day 1 (28) and Day 3 (29). Is 30 > 28 AND 30 > 29? Yes. So, Day 2 is a local maximum.
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Step 3: Look at Day 3 (29). Its neighbours are Day 2 (30) and Day 4 (32). Is 29 > 30? No. So, Day 3 is not a local maximum.
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Step 4: Look at Day 4 (32). Its neighbours are Day 3 (29) and Day 5 (31). Is 32 > 29 AND 32 > 31? Yes. So, Day 4 is a local maximum.
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Step 5: Look at Day 5 (31). Its neighbours are Day 4 (32) and Day 6 (34). Is 31 > 32? No. So, Day 5 is not a local maximum.
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Step 6: Look at Day 6 (34). Its neighbours are Day 5 (31) and Day 7 (33). Is 34 > 31 AND 34 > 33? Yes. So, Day 6 is a local maximum.
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Step 7: Look at Day 7 (33). Only Day 6 (34) is its neighbour. Is 33 > 34? No. So, Day 7 is not a local maximum.
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Answer: The local maximums are on Day 2 (30 degrees), Day 4 (32 degrees), and Day 6 (34 degrees).
Why It Matters
Understanding local maximums helps in finding optimal solutions in various fields. Engineers use it to design structures for maximum strength, and data scientists in AI/ML use it to find the best settings for models. This concept is crucial for careers in data science, engineering, and even medical research for optimizing drug dosages.
Common Mistakes
MISTAKE: Confusing a local maximum with the absolute (global) maximum. | CORRECTION: A local maximum is only the highest point in its immediate neighbourhood, while the absolute maximum is the highest point across the entire function.
MISTAKE: Only checking one neighbour to determine a local maximum. | CORRECTION: You must compare the point's value to ALL its immediate neighbours (both left and right on a graph) to confirm it's a local maximum.
MISTAKE: Assuming a flat section (plateau) cannot contain a local maximum. | CORRECTION: If a function is flat for a segment, any point within that flat segment where the value is equal to its neighbours can be considered a local maximum (or minimum), provided the values on either side are lower.
Practice Questions
Try It Yourself
QUESTION: For the sequence of numbers [5, 8, 6, 10, 9, 12, 7], identify all local maximums. | ANSWER: 8, 10, 12
QUESTION: A mobile network's signal strength (in dBm) was measured every hour: [ -60, -55, -58, -50, -52, -48, -51 ]. Which measurements represent a local maximum (remember, higher dBm is better, so -50 is higher than -60)? | ANSWER: -55, -50, -48
QUESTION: Imagine you are mapping the altitude (in meters) of a trekking path: [100m, 120m, 115m, 130m, 125m, 140m, 140m, 135m]. List all points that are local maximums. Why is 140m appearing twice a local maximum? | ANSWER: 120m, 130m, 140m. The first 140m is a local maximum because it's greater than 125m and equal to the next 140m. The second 140m is a local maximum because it's equal to the first 140m and greater than 135m.
MCQ
Quick Quiz
Which of the following points on a graph represents a local maximum?
A point that is the lowest in its immediate surroundings
A point that is the highest across the entire graph
A point that is higher than all its neighbouring points
A point where the graph is completely flat
The Correct Answer Is:
C
A local maximum is a point whose value is greater than the values of all its immediate neighbours. Option B describes an absolute maximum. Options A and D do not describe a local maximum.
Real World Connection
In the Real World
Local maximums are used in many real-world applications, like in food delivery apps such as Swiggy or Zomato. They use algorithms to find local maximums in demand for delivery riders in certain areas at specific times (e.g., lunch rush in a particular neighbourhood). This helps them position riders efficiently to ensure faster deliveries.
Key Vocabulary
Key Terms
FUNCTION: A relationship where each input has exactly one output | PEAK: The highest point of a hill or mountain; similar to a maximum on a graph | NEIGHBOURING POINTS: Points immediately adjacent to a given point on a graph or sequence | GLOBAL MAXIMUM: The absolute highest value a function reaches over its entire domain
What's Next
What to Learn Next
Now that you understand local maximums, explore 'Local Minimums' to see how we find the lowest points in a region. After that, you can learn about 'Absolute Maximum and Minimums' to find the overall highest and lowest points of a function, which is super useful for optimization problems!
