Min Max

Posted By admin On 30.04.21

Min Maxer
Min Max Thermometer

Min-Max Normalization. Min-max normalization is one of the most common ways to normalize data. For every feature, the minimum value of that feature gets transformed into a 0, the maximum value gets transformed into a 1, and every other value gets transformed into a decimal between 0 and 1. For example, if the minimum value of a feature was 20.
A min/max inventory system is an approach to managing materials or goods in which the business sets a minimum threshold and a maximum level of inventory to hold. When the current supply of an item reaches the minimum level, a new order is placed. When new materials or goods are ordered, the total supply on hand cannot exceed the maximum amount.

You can use Min and Max to determine the smallest and largest values in a field based on the specified aggregation, or grouping. For example, you could use these functions to return the lowest and highest freight cost. If there is no aggregation specified, then the entire table is used.

(verb)
Usually used in the context of roleplaying games, to min/max refers to the act of designing a character in such a way that one minimizes its weaknesses and maximizes its strengths.

Now that I know more about how the gameworks, I'm going to min/max my next character so that it's more effective.

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v. collectively or individually finding someone/something to blame for a problem, as opposed to brainstorming which connotes finding solutions to a problem
n. the act of finding someone/something to blame for a problem
(from a TV ad for HD television)

The committee commenced blamestorming over the situation rather than seeking solutions to the problem.

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verb
1. In a role playing game, to invest most of a player's status points in one or two attributes at the expense of all of the player's other stats.
2. When an individual chooses to develop certain specialties, usually of the intellectual variety, while neglecting other areas of their personal development, usually social skills and charisma.

'Jane min/maxed her barbarian character with level 10 strength but level 0 defense, dexterity, intelligence, charisma, perception and luck. Now she can carry the heaviest axe and attack really hard but she has almost no life, she can't hit the broad side of a barn, can't cast any spells, is socially awkward, oblivious to everything and everyone around her and she can't ever catch a break.'
'John is such an aspie. He's super smart but has no empathy, self-perception or social awareness of any kind. He's intolerable to spend time around. Working in STEM is so draining because guys like him are everywhere and no one holds them accountable for being shitty so they will probably never change.'

by evilspirals November 25, 2015

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Where is a function at a high or low point? Calculus can help!

A maximum is a high point and a minimum is a low point:

In a smoothly changing function a maximum or minimum is always where the function flattens out (except for a saddle point).

Where does it flatten out?Where the slope is zero.

Where is the slope zero?The Derivative tells us!

Let's dive right in with an example:

Example: A ball is thrown in the air. Its height at any time t is given by:

h = 3 + 14t − 5t²

What is its maximum height?

Using derivatives we can find the slope of that function:

h = 0 + 14 − 5(2t)
= 14 − 10t

(See below this example for how we found that derivative.)

Now find when the slope is zero:

10t = 14

The slope is zero at t = 1.4 seconds

And the height at that time is:

h = 3 + 14×1.4 − 5×1.4²

And so:

The maximum height is 12.8 m (at t = 1.4 s)

A Quick Refresher on Derivatives

A derivative basically finds the slope of a function.

In the previous example we took this:

h = 3 + 14t − 5t²

and came up with this derivative:

h = 0 + 14 − 5(2t)
= 14 − 10t

Which tells us the slope of the function at any time t

We used these Derivative Rules:

The slope of a constant value (like 3) is 0
The slope of a line like 2x is 2, so 14t has a slope of 14
A square function like t² has a slope of 2t, so 5t² has a slope of 5(2t)
And then we added them up: 0 + 14 − 5(2t)

How Do We Know it is a Maximum (or Minimum)?

We saw it on the graph! But otherwise ... derivatives come to the rescue again.

Take the derivative of the slope (the second derivative of the original function):

The Derivative of 14 − 10t is −10

This means the slope is continually getting smaller (−10): traveling from left to right the slope starts out positive (the function rises), goes through zero (the flat point), and then the slope becomes negative (the function falls):

A slope that gets smaller (and goes though 0) means a maximum.

This is called the Second Derivative Test

On the graph above I showed the slope before and after, but in practice we do the test at the point where the slope is zero:

Second Derivative Test

When a function's slope is zero at x, and the second derivative at x is:

less than 0, it is a local maximum
greater than 0, it is a local minimum
equal to 0, then the test fails (there may be other ways of finding out though)

'Second Derivative: less than 0 is a maximum, greater than 0 is a minimum'

Example: Find the maxima and minima for:

y = 5x³ + 2x² − 3x

The derivative (slope) is:

y = 15x² + 4x − 3

Which is quadratic with zeros at:

x = −3/5
x = +1/3

Could they be maxima or minima? (Don't look at the graph yet!)

The second derivative is y' = 30x + 4

At x = −3/5:

it is less than 0, so −3/5 is a local maximum

At x = +1/3:

it is greater than 0, so +1/3 is a local minimum

(Now you can look at the graph.)

Words

A high point is called a maximum (plural maxima).

A low point is called a minimum (plural minima).

The general word for maximum or minimum is extremum (plural extrema).

We say local maximum (or minimum) when there may be higher (or lower) points elsewhere but not nearby.

One More Example

Example: Find the maxima and minima for:

y = x³ − 6x² + 12x − 5

The derivative is:

y = 3x² − 12x + 12

Which is quadratic with only one zero at x = 2

Is it a maximum or minimum?

The second derivative is y' = 6x − 12

At x = 2:

it is 0, so the test fails

And here is why:

It is a saddle point ... the slope does become zero, but it is neither a maximum or minimum.

Must Be Differentiable

And there is an important technical point:

The function must be differentiable (the derivative must exist at each point in its domain).

Min Maxer

Example: How about the function f(x) = |x| (absolute value) ?

|x| looks like this:

At x=0 it has a very pointy change!

In fact it is not differentiable there (as shown on the differentiable page).

So we can't use this method for the absolute value function.

Min Max Thermometer

The function must also be continuous, but any function that is differentiable is also continuous, so no need to worry about that.