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Calcatron
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Hello everyone. I'm going to start off by saying that this post may seem like a problem that could be solved with intuition, but I'm a bit of a purist and I like to assure that my ideas are backed up with strong mathematical evidence. I'm studying the Calculus textbook by Thomas and Finney. It's a great book but I'm a bit puzzled at the use of a Calculus theorem. The question is how do we find if a function is concave upward, or concave downward on an interval.
The First Derivative Test for Increasing and Decreasing functions tells us that given that a function is continuous on [a, b] and differentiable on (a, b) then we can apply the following conditionals:
If f' > 0 at each point of (a, b), then f increases on [a, b]
If f' < 0 at each point of (a, b), then f decreases on [a, b]
Here's the precise definition for increasing decreasing functions (you can skip if you already know.)
Let f be a function defined on an interval I and let x1 and x2 be any two points in I.
1. f increases on I if x1 < x2 ==> f(x1) < f(x2)
2. f decreases on I if x1 < x2 ==> f(x2) < f(x1)
So what if we're trying to find whether a function f, differentiable over it's entire domain, is concave up on an interval (x, y). We would take the first derivative, make sure that it's continuous on [x, y] and differentiable on (x, y), then we would see if the second derivative is positive over this interval. At this point we could apply the The First Derivative Test to say that f' increases on the interval [x, y].
But herein lies the problem with certain functions. Suppose we take the first derivative f', but f' is undefined at x, which would imply a discontinuity at it's left endpoint. Then what would we do, could we still apply the The First Derivative Test to the second derivative to see if the first derivative is increasing? I find this difficult to do with mathematical integrity, if you look at the law of Increasing Decreasing functions, it says it only applies if the first function (the one we're trying to find is increasing or decreasing) is continuous on [x, y] and differentiable on (x, y), but our 'first function' the first derivative, tells us that it is not continuous at it's left endpoint!
This observation was based on my analysis of an example where the book assumed everything was 'OKAY' at this point, but I wonder if the book was correct in that assumption and how it was correct if it was.
Any help would be appreciated!
The First Derivative Test for Increasing and Decreasing functions tells us that given that a function is continuous on [a, b] and differentiable on (a, b) then we can apply the following conditionals:
If f' > 0 at each point of (a, b), then f increases on [a, b]
If f' < 0 at each point of (a, b), then f decreases on [a, b]
Here's the precise definition for increasing decreasing functions (you can skip if you already know.)
Let f be a function defined on an interval I and let x1 and x2 be any two points in I.
1. f increases on I if x1 < x2 ==> f(x1) < f(x2)
2. f decreases on I if x1 < x2 ==> f(x2) < f(x1)
So what if we're trying to find whether a function f, differentiable over it's entire domain, is concave up on an interval (x, y). We would take the first derivative, make sure that it's continuous on [x, y] and differentiable on (x, y), then we would see if the second derivative is positive over this interval. At this point we could apply the The First Derivative Test to say that f' increases on the interval [x, y].
But herein lies the problem with certain functions. Suppose we take the first derivative f', but f' is undefined at x, which would imply a discontinuity at it's left endpoint. Then what would we do, could we still apply the The First Derivative Test to the second derivative to see if the first derivative is increasing? I find this difficult to do with mathematical integrity, if you look at the law of Increasing Decreasing functions, it says it only applies if the first function (the one we're trying to find is increasing or decreasing) is continuous on [x, y] and differentiable on (x, y), but our 'first function' the first derivative, tells us that it is not continuous at it's left endpoint!
This observation was based on my analysis of an example where the book assumed everything was 'OKAY' at this point, but I wonder if the book was correct in that assumption and how it was correct if it was.
Any help would be appreciated!