Point of inflection?

A

apple2357

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I have trying to sketch the graph of y= ln(x^2-6x+10). I managed to do this by looking at y=x^2-6x+10 first:

1593245823195.png

Then thinking about y=ln(x) separately:

1593245892701.png

And then building up the ideas to draw y= ln(x^2-6x+10) which turns out to be this:

1593245982211.png

( I also did some calculus to confirm turning points etc.)

However, my question is this:

Neither the quadratic nor the log function have points of inflection but the final graph does. This is a surprise to me. I can explain this algebraically but graphically i can't see why this would be true?

Can anyone help me see why points of inflection appear through a graphical argument?
 
apple2357 said:
I have trying to sketch the graph of y= ln(x^2-6x+10). I managed to do this by looking at y=x^2-6x+10 first:

View attachment 20020

Then thinking about y=ln(x) separately:

View attachment 20021

And then building up the ideas to draw y= ln(x^2-6x+10) which turns out to be this:

View attachment 20022

( I also did some calculus to confirm turning points etc.)

However, my question is this:

Neither the quadratic nor the log function have points of inflection but the final graph does. This is a surprise to me. I can explain this algebraically but graphically i can't see why this would be true?

Can anyone help me see why points of inflection appear through a graphical argument?
Click to expand...
The first graph has a positive "curvature" (constant).

The second graph has a negative curvature - whose "magnitude" is a function of 'x'.

Thus when those are added - the curvature is a function of 'x' and goes through a magnitude of 0.
 
Can you clarify what you mean by positive 'curvature' ? Is that the same as saying the gradient of the graph is always increasing? and lnx the gradient is always decreasing ( as x increases)
 
Last edited:
apple2357 said:
Can you clarify what you mean by positive 'curvature' ? Is that the same as saying the gradient of the graph is always increasing? and lnx the gradient is always decreasing ( as x increases)
Click to expand...
Do you know the equation for 'curvature" of y = f(x)?

If not, google it and let us discuss if you don't understand.
 
Ok, i have picked up a couple of things; the curvature of a function is essentially about how much the function deviates from being a straight line? (And then there is quite a bit about tangent vectors which gets heavy and I get lost in). There is also an intuitive way of thinking about the radius of the circle that fits inside a curve and the curvature is related to this radius, loosely speaking.

So the quadratic has greater 'curvature' than lnx? I can accept that. Not sure what you mean by positive or negative curvature though
 
Don't worry too much about curvature. It should be enough to think about the second derivative. (I'd have to think about whether there's reason to think curvatures would add in composite functions.)

On the other hand, I'm not sure there's an intuitive way to see where the composite function should have its point of inflection; the best I'd expect is to see that it is likely to have one, as the two functions "fight" over the direction of curvature (up or down). But maybe I'm missing something.
 
Dr.Peterson said:
Don't worry too much about curvature. It should be enough to think about the second derivative. (I'd have to think about whether there's reason to think curvatures would add in composite functions.)

On the other hand, I'm not sure there's an intuitive way to see where the composite function should have its point of inflection; the best I'd expect is to see that it is likely to have one, as the two functions "fight" over the direction of curvature (up or down). But maybe I'm missing something.
Click to expand...

Thanks that's helpful. It wasn't so much where the point of inflection arises as much as why it does given two underlying functions that don't have anything of the sort. Your intuitive way of thinking about two functions 'fighting' over the direction is helpful and makes me want to explore other functions! I will have a play on graphical software!
 
You should not expect the composition of functions without inflection points to not have inflection points.

To make it easier, I'll shift your function f(x)= ln [(x-3)^2 +1] to center at 0.

Look at g(x)= ln (x^2 + 1)
x^2 is concave up.
ln x is concave down.

when x is large, the +1 is negligible, so
g ~ ln (x^2) = 2 ln x. So g behaves like ln x, for x large.

when x is small, g = x^2 + ...lower order terms ...
(you can do an expansion near 0 to see this)
So g behaves like x^2 near 0.

Then somewhere in between the concavity changes.
 
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