Why Do Monotonous Functions Have Only One Unique Solution?

[Physicsissuef]

I don't know actually how to know which function is monotonic and which is not. Can you help, please?

 

[tiny-tim]

I don't know actually how to know which function is monotonic and which is not. Can you help, please?

The definition is:

"monotonic increasing" means that, if x < y, then f(x) < f(y).

"monotonic decreasing" means that, if x < y, then f(x) > f(y).

In other words, the graph of a monotonic increasing function always goes up ,
but the graph of a monotonic decreasing function always goes down .​

You can generally tell by differentiating: if the derivative is positive everywhere, then the function obviously increases; if it's negative everywhere, it decreases.

 

[Physicsissuef]

So 5^x+7^x=12^x is not monotonic, because if I get for x=1, y=0. So it doesn't go "upward" nor "backward", right?

 

[trambolin]

You are thinking way too complex for this matter. Just reset your mind and try to visualize...

You have a graph. Think about f(x)=x. Plot it with something and ask this "Does this graph has any chance that after some x, the plot bends and starts to decrease?"

Decreasing means "for some x_1 we have f(x_1) and for some x_2 \geq x_1 in other words, for some x_2 sits on the right hand side of x_1, we have f(x_2), that is less than f(x_1)".

Remember, we are always going towards the positive x by convention... So upward, backward etc. forget about all this. If you save your money and you don't spend it, but once in a while you save some more, the amount can only increase, even if you go from negative(paying borrowed money) to positive(saving!), it is still increasing, which means along the way you can cross zero. that is an example for a monotonic function. But if you cross twice, that means you spend some along the way, thus the amount has decreased somewhere! It is also valid if you are always spending but not saving a dime, thus both directions are valid.

Let me emphasize once again. It is about the functions not about the equations. Equations give you the intersection points of the right and left hand sides,

So, here is another statement for you, the function f(x) = 5^x+7^x-12^x is monotonic between [1,\infty)

What do you think?

 

[Physicsissuef]

I would also say that it is monotonic in (-\infty,1], right?

 

[trambolin]

Almost correct. Be careful, something happens at around 0.5, if you can plot it via some software or even with some calculator you will see it.

 

[Physicsissuef]

yes,
x=-2, y=0.0534
x=-1, y=0.259
x=0,5, y=4.88

x=1 , y=0

so it is monotonic increasing for <br /> (-\infty,1)<br />
And how will I know from which point to which one is monotonic increasing, and decreasing? Should I look first for the 0?

 

[trambolin]

First of all, you cannot write (-\infty,1), I understand that you are trying to say somewhere before 1. But what you write is all points from negative infinity to 1 excluding only 1. These are very dangerous mistakes, that you cannot do even when you are asleep!

Regarding your question, just check your notes for finding maxima and minima of functions. I think that is a fair hint for it.

 

[Physicsissuef]

First of all, you cannot write (-\infty,1), I understand that you are trying to say somewhere before 1. But what you write is all points from negative infinity to 1 excluding only 1. These are very dangerous mistakes, that you cannot do even when you are asleep!

Regarding your question, just check your notes for finding maxima and minima of functions. I think that is a fair hint for it.

on this fuction minima is x=1, but maxima I don't know... Why (-\infty,1) (moonotonic decreasing) is not correct?