Posted by amy on November 13, 2002 at 18:30:07:
In Reply to: Calculus I Help posted by ashesover on November 13, 2002 at 14:24:50:
: someone help me determine the absolutes, my book says if it is at an endpoint on an interval then it is not an absolute but a local
: also the second derivative is just a line, how can you prove the concavity?
To find the absolutes you need to determine first the critical points (wherever the first derivative doesn't exist or is equal to zero). Since we are given a closed interval, the endpoints are also critical points. They are local minima or maxima, and possibly, but not definitely absolute. To find out if a critical value is an absolute you plug that value into the second derivative. If you get a negative number it is an absolute maximum, meaning it is concave down, if you get a positive number it is an absoluted minimum, meaning it is concave up (See second derivitive test in your book).
Also once you find inflection points you can use these intervals with the derivative to determine concavity. take an X value somewhere in the interval (any) and plug it in to the first derivative. whether this number is positive or negative tells you whether it is increasing or decreasing on that interval. When you look at each interval like this you can determine concavity. For instance if you have that on [0,1] the function is increasing (the derivative is positive on some point between 0 and 1) and on [1,2] the function is decreasing (the derivative is negative on some point between 1 and 2), then the function is concave down on [0,2].
keep in mind those actual numbers have nothing to do with your particular problem, but only serve to illustrate what I'm talking about.
good luck!