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Minimum: Definition and Example | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">MinimumMinimum: Definition and ExampleTable of ContentsUnderstanding Minimum in Mathematics Definition

In mathematics, a minimum is the smallest value in a set of numbers or the lowest point on a graph of a function. When we find a minimum, we're looking for the smallest number in a group or the lowest point a curve reaches. For example, in the set {7, 3, 9, 5, 2}, the minimum value is 2 because it's smaller than all other numbers in the set. In everyday life, we use minimums when we need to know the lowest temperature of the day, the shortest student in a class, or the smallest number of points needed to win a game.

There are different types of minimums we might study. A global minimum (or absolute minimum) is the lowest value across an entire set or function. A local minimum (or relative minimum) is a value that's lower than all nearby values, but might not be the lowest overall. For functions, a minimum can occur at a point where the slope changes from negative to positive, which is a turning point on a graph. Finding minimums is important in many areas of math and helps us solve real-world problems like finding the best price, the most efficient strategy, or the optimal dimensions for an object.

Examples of Minimum in Mathematics Example 1: Finding the Minimum Value in a Set of Numbers Problem:

Find the minimum value in this set of numbers: 12,7,15,9,4,1112, 7, 15, 9, 4, 1112,7,15,9,4,11.

Step-by-step solution:

Step 1, Write down all the numbers so you can see them clearly.

12,7,15,9,4,1112, 7, 15, 9, 4, 1112,7,15,9,4,11

Step 2, Start by picking any number as your "current minimum."

Let's start with the first number: 121212 Current minimum = 121212

Step 3, Compare this minimum with the next number in the list.

Next number is 777. Is 777 smaller than 121212? Yes, it is! So our new minimum is 777.

Step 4, Continue comparing with each number in the list.

Next number is 151515.

Is 151515 smaller than 777? No, it's larger.

Minimum is still 777.

Next number is 999.

Is 999 smaller than 777? No, it's larger.

Minimum is still 777.

Next number is 444.

Is 444 smaller than 777? Yes, it is!

So our new minimum is 444.

Next number is 111111.

Is 111111 smaller than 444? No, it's larger.

Minimum is still 444.

Step 5, After checking all numbers, we can give our answer.

The minimum value in the set {12,7,15,9,4,1112, 7, 15, 9, 4, 1112,7,15,9,4,11} is 444. Example 2: Finding the Minimum Height of a Ball Problem:

A ball is thrown upward with an initial velocity of 202020 meters per second from a height of 555 meters. The height of the ball (in meters) at time t seconds is given by the formula h=5+20t−5t2h = 5 + 20t - 5t^2h=5+20t−5t2. What is the minimum height the ball reaches if it is caught when it returns to the initial height?

Step-by-step solution:

Step 1, Understand the problem context.

The ball starts at height 555 meters It is thrown upward with initial velocity 202020 m/s We need to find the minimum height during its flight The ball is caught when it returns to 555 meters

Step 2, Find the times when the ball is at the initial height of 555 meters.

Set h=5h = 5h=5: 5+20t−5t2=55 + 20t - 5t^2 = 55+20t−5t2=5 20t−5t2=020t - 5t^2 = 020t−5t2=0 5t(4−t)=05t(4 - t) = 05t(4−t)=0 So t=0t = 0t=0 or t=4t = 4t=4 The ball starts at t=0t = 0t=0 and returns to the initial height at t=4t = 4t=4 seconds.

Step 3, Examine what happens between these times.

The ball goes up and then comes down To find when the ball reaches maximum height (not minimum), we can find when velocity is zero The velocity is dhdt=20−10t\frac{dh}{dt} = 20 - 10tdtdh​=20−10t Setting this equal to zero: 20−10t=020 - 10t = 020−10t=0 Solving: t=2t = 2t=2 seconds This is when the ball reaches its highest point

Step 4, Determine the minimum height during the flight.

Since the ball is thrown upward from 555 meters, reaches a maximum, then falls back to 555 meters The height at t=0t = 0t=0 and t=4t = 4t=4 is 555 meters The height at any time between 000 and 444 is higher than 555 meters Therefore, the minimum height during the ball's flight is 555 meters

Step 5, State the final answer.

The minimum height the ball reaches is 555 meters. Example 3: Finding the Minimum Value of a Mathematical Expression Problem:

Find the minimum value of y=x2−6x+10y = x^2 - 6x + 10y=x2−6x+10 for all real values of xxx.

Step-by-step solution:

Step 1, Understand what we're looking for.

We need to find the value of xxx that gives us the smallest possible value for yyy.

Step 2, Find where the slope of the function changes from negative to positive.

The slope is given by the derivative: dydx=2x−6\frac{dy}{dx} = 2x - 6dxdy​=2x−6

Step 3, Set the derivative equal to zero to find where the slope changes.

2x−6=02x - 6 = 02x−6=0 2x=62x = 62x=6 x=3x = 3x=3

Step 4, Check that this is a minimum (not a maximum) by making sure the second derivative is positive.

The second derivative is d2ydx2=2\frac{d^2y}{dx^2} = 2dx2d2y​=2, which is positive. This confirms we've found a minimum.

Step 5, Calculate the minimum value by plugging x=3x = 3x=3 into the original function.

y=x2−6x+10y = x^2 - 6x + 10y=x2−6x+10 y=(3)2−6(3)+10y = (3)^2 - 6(3) + 10y=(3)2−6(3)+10 y=9−18+10y = 9 - 18 + 10y=9−18+10 y=1y = 1y=1

Step 6, State the final answer.

The minimum value of y=x2−6x+10y = x^2 - 6x + 10y=x2−6x+10 is 111, which occurs when x=3x = 3x=3. Comments(3)NNurseBethNovember 5, 2025I've used this minimum definition to teach my students. It's clear and the examples really help them grasp the concept.

JJournalistRachelNovember 4, 2025I've used this minimum def to help my students. It's clear and the examples make it easy for them to grasp. Great resource!

MCMs. CarterSeptember 17, 2025I used the 'Minimum' definition and examples to help my kids understand optimization problems in their homework. It’s clear and easy to follow—such a great resource for both parents and teachers!

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