What is a Relative Minimum in Calculus?

Learn about the relative minimum in calculus and how to find it along with when you might use the knowledge. Read further here!

It’s easy to visualize the minimum and maximum values of a function since the minimum is the lowest point on a graph while the maximum is the highest. If you’re looking at a graph with points on it, just choose the lowest one. You can find four possible values of a function on a graph including:

  • Absolute maximum
  • Relative maximum
  • Absolute minimum
  • Relative minimum

For the purpose of this article, we want to look at the relative minimum. The relative minimum won’t be the lowest point on the graph but a relative interval between the lowest and highest points. For instance, assume we have a graph with an x and y-axis. Let’s put three points five points on the chart that crisscross x as demonstrated.

The first three points to the left of y end up being -9, 11, and -6. To the right of y, we have two points at 13 and -8. In this very simple scenario, the relative minimum is -6 and the relative maximum is 11 because they fall within the domain of x. The domain might get a little tricky at times, but it’s all simple if you think of it as merely being all the variables that make the function work.

Understanding the Domain of a Function

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The domain of x could be a simple set of numbers beginning, from left to right, with negative six and ending with positive six. However, we don’t have to use the entire domain. Your function could end up confined between negative four and positive four. That’s how we end up with relative minimums and relative maximums. Other points may exist, obviously, but they don’t fall into the correct domain.

Let’s look at a simple equation like y is equal to the square root of x plus four. The domain of this equation is illustrated with the function x is greater than or equal to negative four. We can’t go any lower than negative four of the equation breaks down because the denominator of a fraction can’t be a zero and number under our square root sign is positive and must remain that way.

If you try to solve this using a number lower than negative four, it won’t work or produce an answer. Try it using -11 or even -5, and the result will always be incorrect. The domain of a function is a little more complicated than our simple scenario here, but to understand the relative minimum this definition is all you need at the moment.

My Relative Minimum is Missing

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Sometimes on a test, you may have a problem that asked you to solve the equation x is equal to x squared using only two intervals like negative one and positive two. If you graph that, the relative and absolute minimum are both stuck on zero. The negative one can’t be the relative minimum because it’s at the end of our interval.

In cases like this, the answer is always one of several multiple-choice solutions. The correct answer here would likely be none of the above. Passing a math test and math don’t always go together well, and little tricks like this get put on the test to make sure you’re paying attention not guessing the answer when you can’t solve the problem.

Revisiting Relative Minimum

If you didn’t notice, the first explanation of relative minimum is overly complicated and an almost robotic way of explaining it. You need to understand how to find the relative minimum using the most complicated methods possible if you want to excel at calculus. However, an easier way to find the relative maximum or minimum exists, if you graph everything.

Assuming your graph has a line with a starting and ending point, and more than two points, the relative minimum is the lowest part of your line where it changes direction. The opposite is true for the relative maximum. If your line starts and negative eleven and moves to a positive number then back to negative six before climbing again, the negative six if where the direction changes and the relative minimum.

Why Do I Need to Learn This?

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Well, for one, you need to know this to pass your algebra class. That seems like an excellent reason. Aside from that, many jobs require strong math skills and they pay well. If you check the Bureau of Labor Statistics website, you can look up your dream job and learn more about it. You can also use that website to look into other positions, find out what they pay, and possibly find a new career.

However, many of the highest paying jobs require strong math skills. Understanding how to find the relative minimum or maximum is essential in several positions revolving around economics and marketing. However, the terms used in each industry may differ, and it’s unlikely your boss will ever email a spreadsheet and ask you to find the relative minimum of anything on it.

You may be surprised to learn that some jobs you thought might be easy once you got the right degree actually require a high level of math knowledge. For instance:

  • Robotics engineers use a lot of math
  • Pilots that fly jets or commercial airliners must have a good grasp on math
  • Game designers need trigonometry to make things move and algebra to make the blow-up
  • Animators fall into the same category as game designers
  • Sports announcers use much more math than you can imagine

Many countries require pilots to take numerical reasoning tests, and they don’t allow calculators. Take a practice test and see how well you do on it. While this isn’t a direct instance of you needing to derive the relative minimum of a function, it’s an excellent example of how vital understanding math might be to your academic and future careers.

While they may call it by another name, several highly sought and well-paying jobs use calculus heavily. Let’s look at a couple of those with a little more detail starting with a mathematician job. They don’t just teach math, and many get employed at places like NASA. Check out NASA’s basic education requirements for some of their more important and fun jobs.

A cryptographer is another excellent example of an almost a pure math-based job. They work everywhere these days from private companies that develop privacy products to governments that need to secure things for transmission. They also work at places like the NSA where they try to secure items and decipher elements that may be encrypted.

Economists work with market data, statistics, and models to figure out current and future economic trends. Highs and lows on a graph might be their entire workload sometimes. Many of them work for various government agencies, but economist jobs in the private sector may become more popular in the future.

People that work with investments like analysts and bankers need a lot of math to do their jobs well. Seeing highs and lows on a graph might be the only way they get to make a decision. Their decisions might cost someone a lot of money if they get the math wrong or don’t understand the graph.

Accountants probably use math as often as a rocket designer. Math is practically all they do. They do things like figure out payroll for companies and prepare taxes for people. A mistake on their end could mean your paycheck is wrong or you end up paying the government back more than you owe. The opposite effects are possible as well.

Geodesists tell us things like how far it is to celestial bodies like the sun or distant galaxies. Sometimes these objects move, and the distance may fluctuate which results in maximum and minimum distance measurements. They have to factor in many things mathematically like gravity and mass to get an accurate measurement.

Mathematics modelers do several things including creating complex computer simulations and investigating number theories. If we need to know whether or not a comet will hit us, these people develop models based on data from various sources, including geodesists, to determine the chances of a comet smacking into the planet. That’s high on our list of essential jobs using math.

Many of the higher paying versions of these jobs require a doctorate to compete in the job market.

Some Final Notes

Granted, out definitions and problems lean toward the simpler side of finding the relative minimum, but you should understand how it works from a broad viewpoint now. If you need more help, you can find plenty of videos online that explain how to arrive at the relative minimum and how it may apply to other pre-calculus and calculus problems.

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