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The first derivative will tell you where the slope of the function is 0. This is a prerequisite for an inflection point.
The other requirement is that the function change `directions' from one side of the point to the other.
In order for that to happen, the slope must go from being negative to being positive or positive to negative. The second derivative can tell us whether or not that is happening.
Sorry, that's completely wrong.
Started explaining the wrong concept.
Let's start from scratch! An inflection point is where a function changes from being concave upwards to being concave downwards.
Functions that are concave upwards have a slope that is increasing (either becoming more positive or less negative), and those that are concave downwards have a slope that is decreasing (either becoming less positive or more negative).
Remember the inflection point tells us where the concavity of the underlying function changes from concave up to concave down.. So tracing this through the first derivative....an inflection point (on the original function) becomes a max or min on the first derivative function and therefore a zero on the second derivative function. That is why it is easiest to identify the inflection points from the second derivative....they are the zeros!
To convince yourself of this, you can do a quick sketch of a regular parabola, which is concave up, and sketch tangent lines at the edges and in the middle of the curvatures. You should be able to see that the tangent lines go from a very negative slope to a mid-negative slope to a zero slope to a mid-positive slope to a very positive slope. You can do the same with a parabola flipped over the y axis to convince yourself of how concave down implies a tangent slope that is decreasing. Then yes, apply descartes's information :)
thank you both! But I guess I am confused with what the second derivative IS.
I know that the first derivative is the expression for the slope of the tangent line; therefore, when it is zero, and there is a change in sign, we can find the extrema. But why is it when the second derivative=0, we find the inflection point? descartes sort of lost me on tracing through the first derivative...
So, the first derivative describes the slope of the function at any given point.
Basically it's a visualization (or description) of how the slope changes.
Similarly, the second derivative describes the slope of the *first* derivative at any given point.
i depende on how the slope is
thanks. I am still sort of muddy. Descartes said, "an inflection point (on the original function) becomes a max or min on the first derivative function and therefore a zero on the second derivative function. "
why is that?
In terms of the first function, it describes how fast and in what direction the function's slope is changing.
So, as I was saying earlier, the inflection point is when the function changes from concave up to down or vice versa.
This means that the *slope* of the function changes from *increasing* to *decreasing* or the other way around.
This means that the slope of the first *derivative* changes from *positive* to *negative* or vice versa.
Which means that the *value* of the *second* derivative changes from positive to negative or vice versa.
And between positive and negative is zero :)
that can make a diffrence in the question itself so the slop might be *increseing* or *decreseung* ethir way
Does that make a little more sense?
'slope* of the function changes from *increasing* to *decreasing* or the other way around"
do you mean the original function?
isn't that the extrema though?
wait. I think I get it...
I mean the original function, yes. And the extrema are where the slope of the original function changes from positive to negative, not where it changes from actually increasing to decreasing.
:( what is the difference? I'm sorry.
Increasing is a description of the *rate* at which something is changing. So if the slope of the original function is increasing, its slope is getting more positive *faster*. Instead of just getting more positive.