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What is a Local Maximum on a Graph?
Grade Level:
Class 9
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
A local maximum on a graph is a point where the value of the function is higher than all the points around it in its immediate neighbourhood. Think of it as the peak of a small hill or a bump on a roller coaster track, even if there are higher peaks elsewhere on the graph.
Simple Example
Quick Example
Imagine you're tracking the number of runs Virat Kohli scores in different matches over a season. If in one match he scores 95 runs, and in the matches just before and after that one, he scored 60 and 70 runs respectively, then 95 runs is a local maximum for that small series of matches. It's the highest point for that particular 'hill' of scores.
Worked Example
Step-by-Step
Let's look at the daily temperature in Celsius for a week in Delhi:
Day 1: 28°C
Day 2: 30°C
Day 3: 32°C
Day 4: 31°C
Day 5: 29°C
Day 6: 30°C
Day 7: 27°C
Step 1: Identify the temperatures for each day.
---Step 2: Look at Day 3 (32°C). Compare it with its neighbours: Day 2 (30°C) and Day 4 (31°C).
---Step 3: Is 32°C greater than 30°C? Yes. Is 32°C greater than 31°C? Yes.
---Step 4: Since 32°C is higher than both its immediate neighbours, Day 3 represents a local maximum.
---Step 5: Look at Day 6 (30°C). Compare it with its neighbours: Day 5 (29°C) and Day 7 (27°C).
---Step 6: Is 30°C greater than 29°C? Yes. Is 30°C greater than 27°C? Yes.
---Step 7: Since 30°C is higher than both its immediate neighbours, Day 6 also represents a local maximum.
Answer: The local maximum temperatures are 32°C (on Day 3) and 30°C (on Day 6).
Why It Matters
Understanding local maximum helps in finding optimal points or peaks in data, which is crucial in fields like AI/ML for training models, or in economics to find the highest profit points. Engineers use it to design systems that perform best, and data scientists use it to spot trends and make predictions.
Common Mistakes
MISTAKE: Confusing a local maximum with the absolute maximum (the highest point on the entire graph). | CORRECTION: A local maximum is only the highest in its *immediate neighbourhood*, not necessarily the highest point overall.
MISTAKE: Thinking any point that is simply 'high' is a local maximum. | CORRECTION: A point must be higher than *both* its left and right neighbours (or surrounding points in 2D/3D) to be a local maximum.
MISTAKE: Only checking one neighbour (e.g., just the point to the left). | CORRECTION: Always check *all* immediate neighbours to confirm if a point is truly higher than everything around it.
Practice Questions
Try It Yourself
QUESTION: A student's scores in 5 consecutive tests were 75, 80, 78, 85, 82. Identify the local maximum scores. | ANSWER: 80 (because 80 > 75 and 80 > 78) and 85 (because 85 > 78 and 85 > 82).
QUESTION: The height of a ball thrown in the air at different times is recorded as: 0m, 5m, 8m, 7m, 3m, 0m. What is the local maximum height? | ANSWER: 8m (at 8m, its neighbours are 5m and 7m, and 8 > 5 and 8 > 7).
QUESTION: The daily sales (in thousands of rupees) of a roadside chai stall for 7 days were: 10, 12, 11, 9, 13, 10, 8. What are the local maximum sales figures? Is there an absolute maximum? | ANSWER: Local maximums are 12 (because 12 > 10 and 12 > 11) and 13 (because 13 > 9 and 13 > 10). The absolute maximum is 13.
MCQ
Quick Quiz
Which of the following points on a graph represents a local maximum?
A point that is the lowest in its immediate area.
A point that is higher than all other points on the entire graph.
A point that is higher than its immediate neighbouring points.
A point where the graph is flat.
The Correct Answer Is:
C
A local maximum is a point that is higher than its immediate neighbours. Option B describes an 'absolute maximum', not just a local one. Options A and D are incorrect descriptions of a maximum.
Real World Connection
In the Real World
Local maximums are used in many real-world applications. For example, when you use a navigation app like Google Maps or Ola, it often calculates the 'peak traffic' hours (a local maximum in traffic density) to suggest the best route. In stock markets, analysts look for local maximums in stock prices to identify resistance levels or potential selling points. Even in cricket analytics, finding a local maximum in a player's strike rate can show their peak performance period.
Key Vocabulary
Key Terms
PEAK: The highest point of a hill or curve | NEIGHBOURHOOD: The immediate surrounding points of a specific point on a graph | ABSOLUTE MAXIMUM: The single highest point on an entire graph | FUNCTION: A relationship where each input has exactly one output
What's Next
What to Learn Next
Great job understanding local maximums! Next, you should explore 'What is a Local Minimum on a Graph?'. It's the opposite concept and will help you fully understand how to identify all the important turning points on any graph or data set.
