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## Differential Calculus

### Course: Differential Calculus>Unit 5

Lesson 5: Absolute (global) extrema

# Absolute minima & maxima review

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.3 (EK)
Review how we use differential calculus to find absolute extremum (minimum and maximum) points.

## How do I find absolute minimum & maximum points with differential calculus?

An absolute maximum point is a point where the function obtains its greatest possible value. Similarly, an absolute minimum point is a point where the function obtains its least possible value.
Supposing you already know how to find relative minima & maxima, finding absolute extremum points involves one more step: considering the ends in both directions.

## Finding absolute extrema on a closed interval

Extreme value theorem tells us that a continuous function must obtain absolute minimum and maximum values on a closed interval. These extreme values are obtained, either on a relative extremum point within the interval, or on the endpoints of the interval.
Let's find, for example, the absolute extrema of h, left parenthesis, x, right parenthesis, equals, 2, x, cubed, plus, 3, x, squared, minus, 12, x over the interval minus, 3, is less than or equal to, x, is less than or equal to, 3.
h, prime, left parenthesis, x, right parenthesis, equals, 6, left parenthesis, x, plus, 2, right parenthesis, left parenthesis, x, minus, 1, right parenthesis, so our critical points are x, equals, minus, 2 and x, equals, 1. They divide the closed interval minus, 3, is less than or equal to, x, is less than or equal to, 3 into three parts:
Intervalx-valueh, prime, left parenthesis, x, right parenthesisVerdict
minus, 3, is less than, x, is less than, minus, 2x, equals, minus, start fraction, 5, divided by, 2, end fractionh, prime, left parenthesis, minus, start fraction, 5, divided by, 2, end fraction, right parenthesis, equals, start fraction, 21, divided by, 2, end fraction, is greater than, 0h is increasing \nearrow
minus, 2, is less than, x, is less than, 1x, equals, 0h, prime, left parenthesis, 0, right parenthesis, equals, minus, 12, is less than, 0h is decreasing \searrow
1, is less than, x, is less than, 3x, equals, 2h, prime, left parenthesis, 2, right parenthesis, equals, 24, is greater than, 0h is increasing \nearrow
Now we look at the critical points and the endpoints of the interval:
xh, left parenthesis, x, right parenthesisBeforeAfterVerdict
minus, 39minus\nearrowMinimum
minus, 220\nearrow\searrowMaximum
1minus, 7\searrow\nearrowMinimum
345\nearrowminusMaximum
On the closed interval minus, 3, is less than or equal to, x, is less than or equal to, 3, the points left parenthesis, minus, 3, comma, 9, right parenthesis and left parenthesis, 1, comma, minus, 7, right parenthesis are relative minima and the points left parenthesis, minus, 2, comma, 20, right parenthesis and left parenthesis, 3, comma, 45, right parenthesis are relative maxima.
left parenthesis, 1, comma, minus, 7, right parenthesis is the lowest relative minimum, so it's the absolute minimum point, and left parenthesis, 3, comma, 45, right parenthesis is the largest relative maximum, so it's the absolute maximum point.
Notice that the absolute minimum value is obtained within the interval and the absolute maximum value is obtained on an endpoint.
Problem 1
• Current
f, left parenthesis, x, right parenthesis, equals, x, cubed, minus, 3, x, squared, plus, 12
What is the absolute maximum value of f over the closed interval open bracket, minus, 2, comma, 4, close bracket?

Want to try more problems like this? Check out this exercise.

## Finding absolute extrema on entire domain

Not all functions have an absolute maximum or minimum value on their entire domain. For example, the linear function f, left parenthesis, x, right parenthesis, equals, x doesn't have an absolute minimum or maximum (it can be as low or as high as we want).
However, some functions do have an absolute extremum on their entire domain. Let's analyze, for example, the function g, left parenthesis, x, right parenthesis, equals, x, e, start superscript, 3, x, end superscript.
g, prime, left parenthesis, x, right parenthesis, equals, e, start superscript, 3, x, end superscript, left parenthesis, 1, plus, 3, x, right parenthesis, so our only critical point is x, equals, minus, start fraction, 1, divided by, 3, end fraction.
Intervalx-valueg, prime, left parenthesis, x, right parenthesisVerdict
left parenthesis, minus, infinity, comma, minus, start fraction, 1, divided by, 3, end fraction, right parenthesisx, equals, minus, 1g, prime, left parenthesis, minus, 1, right parenthesis, equals, minus, start fraction, 2, divided by, e, cubed, end fraction, is less than, 0g is decreasing \searrow
left parenthesis, minus, start fraction, 1, divided by, 3, end fraction, comma, infinity, right parenthesisx, equals, 0g, prime, left parenthesis, 0, right parenthesis, equals, 1, is greater than, 0g is increasing \nearrow
Let's imagine ourselves walking on the graph of g, starting all the way to the left (from minus, infinity) and going all the way to the right (until plus, infinity).
We will start by going down and down until we reach x, equals, minus, start fraction, 1, divided by, 3, end fraction. Then, we will be forever going up. So g has an absolute minimum point at x, equals, minus, start fraction, 1, divided by, 3, end fraction. The function doesn't have an absolute maximum value.
Problem 1
• Current
g, left parenthesis, x, right parenthesis, equals, start fraction, natural log, left parenthesis, x, right parenthesis, divided by, x, end fraction
What is the absolute maximum value of g ?

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• Please, tell me What is the meaning of before and after in the above table and also whats the meaning of that table... Please Help me!
(1 vote)
• The "before" simply refers to whether the function is increasing or decreasing BEFORE the x-value in question (e.g. -3). The "after" refers to whether the function is increasing or decreasing AFTER the x-value in question. The places where you see a dash (or what looks like a minus sign) are where it doesn't matter because the x-value is an endpoint (e.g. for -3 and +3, since these are the endpoints, what happens before -3 and what happens after +3 are irrelevant).
• Does x^3 - 3x^2 + 12 have a global maximum? For my sign chart, it shows that the function continues to increase as x approaches positive and negative infinity. How do I determine if there are global extrema based on the sign chart?
(1 vote)
• If the function continues to increase as x → +∞ (or -∞) then there can never be a maximum value and therefore no global maximum.

This is definitely true for x³ - 3x² + 12 and I believe will be true for any single variable polynomial with an odd degree ... in fact I don't think any such polynomial could ever have any global extrema (maximum or minimum) ...
• Why is this function defined for x=0?
• When we are looking at the critical points and the endpoints to compare them the table has a before and after columns. What is the before and after referring to? Before and after what?
• We need to determine whether the slope is positive or negative before and after the critical points and the end points. Hence, the comparison table helps to determine this.
• If the absolute value of a graph falls on a removable discontinuity or a hole does it exist? The function value does not exist as it is an open dot, but can it still be considered the absolute max/min?
(1 vote)
• No. In that case, there would be no extremum on that particular interval containing the discontinuity.

However, a special case can be made for something like f(x) = x^2 if x ≠ 0, -1 if x = 0, where a relative minimum does exist. So in general, if a function is undefined somewhere, you should still check for extrema.
• Thank you for this helpful summary.

I noticed that when finding global extrema on a closed interval, once you know the critical points and endpoints that are candidates for being global extrema, you just have to plug those values into the original function along with the endpoints on the closed interval. You do not have to actually evaluate the derivative in between each critical point (although this would further inform your answer from an intuitive point of view).

My question is whether it is completely necessary to always evaluate the derivative in between critical and end points 'when finding global extrema' or whether you can just rely on the output of those values when plugged into the initial function. (This way, the critical/end point with the highest output for the original function is the global maximum... etc.) I think this only works on closed intervals with endpoints.

• what if the lowest value within the interval is a relative maximum, wouldn't that be the abs. minimum of the interval? because "absolute minimum" is defined to be the "lowest possible value" within something.
(1 vote)
• If a point is a relative maximum, that means there are nearby points with lower y-values, so the absolute minimum cannot be a relative maximum. Likewise, the absolute maximum cannot be a relative minimum.