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I am having trouble making sense of how to figure out what a graph looks like just by knowing the critical numbers and the intervals the function is increasing and when it is decreasing. For example I have $(-\infty, -1), (0,1)$ as negative and $(-1,0), (1,+\infty)$ as positive.

I know that the it can only change on critical numbers so that means if I have increase, critical number and then decreasing then I have a graph that is going up and then down and the critical number is a minimum. If I have decrease, critical number and then an increase I have a minimum.

The problem I ma having is determing how to view these. If I start from the left of the function I get concave up (minimums) with decrease, critical number, increase.

If I do the same with an increase, critical number decrease by starting from the right side I get a concave down (max).

How do I know which is correct when working with just this information? It seems to depend entirely on which side I start on.

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I think that your question is not very clear - What is the function you are having trouble with? – NoChance Oct 13 '11 at 23:26
Any function really, it does not matter. I used $f(x)= x^4 -2x^2+3$ I then found the derivative and then I found the criticals numbers, -1, 0, 1. I then found the intervals where it is increasing and decreasing as noted in the original post. This is where I am having trouble telling what is a local minimum and maximum. – user138246 Oct 13 '11 at 23:31
up vote 2 down vote accepted

The sign of your function seems to change at $-1$, $0$ and $1$ rather than the direction.

You have not specified much more about your function. If it is continuous then it will have at least one local maximum in $(-1,0)$ and the direction will change there, and similarly at least one local minimum in $(0,1)$. This will not let you conclude much about where it will be concave or convex (apart from being concave up close to a local minimum etc.).

I would have thought each of the red and blue functions below would meet your specifications:

enter image description here

If I have misunderstood and it is the direction which changes then you still cannot conclude anything about concavity and convexity, though there will be local minima at $-1$ and $1$ (at least one of which will be a global minimum) and a local maximum at $0$

enter image description here

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Take a look at: Link-1 for a good explanation of the concept.

Take a look at the graph below.enter image description here

Your critical numbers are -1, 0, 1 are shown.

To determine your local min/max use the rule shown below.

Second derivative is = ${f}''(x)=12 x ^{2} - 4 $

Second derivative sign at critical points:

at -1 --> (+) --> local min.

at 0 --> (-) --> local max.

at 1 --> (+) --> local max.

enter image description here

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