Question

How do I identify the "extreme" values of the graph of a function?

Answer 1

Specifically for the graph of\[f(x)=x ^{4}-4x ^{3}-x ^{2}+12x-2\]

Answer 2

Which I already graphed... I just don't know what "extreme" values are.

Answer 3

Extreme values are the so-called "max"'s and "min"'s that you may have heard of before. Those are the spots on the graph where you hit the peak, sort of like the top of a hill. It may be asking you for absolute extrema, which is just the highest point in terms of the whole graph, while local extrema are just the mini hills. To find these, you want to find the derivative of the equation and set it equal to 0. The reason you set it to 0 is because the value of the derivative will transition to negative when the original graph starts to change direction (either stopping going downwards and going up, or stopping going upwards and going down). The exact moment this happens is right in between the positive and negative areas of the derivative's graph, which is 0. There may be more than one spot this happens at, in which case all of the spots or local extrema. If the derivative transitions from negative to positive, it is a local minimum, and positive to negative is a maximum.

Answer 4

By the way a minimum looks like an upside-down hill, which I forgot to mention.

Answer 5

Alright, thank you (: Not quite sure how to find the derivative but I'm looking through my notes

Answer 6

Okay yeah so I can't find that in my notes either ._.

Answer 7

use the exponent rule ( I don't know in english, I'm brazilian )

df(x)/dx = 4x^3-4*3xˆ2-2x+12

Answer 8

Are you in pre-calc?

Answer 9

Yes I'm in pre-calc

Answer 10

If you have not been taught calculus yet then you can find the extreme values as follows:
You said you already graphed the function. Look for the points where it has peaks and valleys. Those are your relative maximum and minimum respectively. Just note down the x, y coordinates and those are the points of extrema. The y value will be the max or min and the x value will be where the function attains max or min.

Answer 11

okay (: thank you!

Answer 12

you are welcome.
Could you tell me what are (x,y) values you got from the graph?

Answer 13

Well the two "valleys" are at about (-1, -10) and (2.75, -2.25). And the peak is at about (1, 6)... what exactly do I do with those? Do I use all three of them?

Answer 14

Very nice! Pretty close. If you want more accurate values you can either use a graphing calculator or online sites such as: https://www.desmos.com/calculator

At two decimal accuracy the points are: (-0.94, -10.06) ; (1.14, 6.14) ; (2.79, -2.58)

Answer 15

The middle one is the maximum and the other two are minima. Yes those three points represent the relative extrema. The extreme values are the y-coordinates. The x-coordinates are where the extreme value occurs.