Question

How do I find the absolute maximum and minimum of the function, if I am just given a function f(x), without the graph?

Answer 1

(I know how to derive)

Answer 2

Ah my first question^

First step is to take the first derivative of the function...then set that equal to 0

This will give you "critical points" which are potential maxima and minima

Answer 3

yes, I know (critical points are also those at which f(x) doesn't exist. )

Answer 4

And how do I know which of the critical points is the abs max and min ?

Answer 5

Indeed^ Following this step we take the second derivative of the function

The second derivative at 'x' will give you which of those points are maxima or minima (or a saddle point)

If the second derivative turns out negative we have a minimum
If it turns out negative, we have a maximum

Answer 6

I don't really get it. Can I do an example and you correct me if anything. Is that alright?

Answer 7

No problem!

Answer 8

Lets say I have f(x)=x^4+x^3-2x+1

Answer 9

bad example

Answer 10

I;ll get an example and do it in another question. I;ll get the book. sorry abt that

Answer 11

tnx for the help

Answer 12

Anytime

Answer 13

But for reference, we can do that example

x^4 + x^3 - 2x + 1

First derivative would be

4x^3 + 3x^2 - 2

Set that = to 0

4x^3 + 3x^2 - 2 = 0
4x^3 + 3x^2 = 2
x(4x^2 + 3x) = 2

I'll just skip the process of completing the square here and all that and I get x is roughly equal to 0.60701

That is our 1 and only critical point that I found here

So we just take the second derivative now...and plug in That for 'x'

second derivative would be

12x^2 + 6x

Plugging in x = 0.60701 we get a positive number...thus...a minimum