knowLittle said:Homework Statement
y= (x^2 -7) e^x
The Attempt at a Solution
I'm trying to find inflection points by setting the second derivative=0
I found that the derivative is:
##2xe^{x}+x^{2}e^{x}-7e^{x}=0##
##e^{x}[2x+x^{2}-7]=0##
Then, the 2nd derivative:
##e^{x}[(x-1)(x+5)]=0##, then the inflection points are at x=- infinity; 1; -5.
Where ##e^{x}=0 ##, happens when x=-infinity
Is it correct to use - infinity as a value of x ?
The purpose of finding these intervals is to better understand the behavior and shape of a given function. Knowing whether a function is convex or concave can help in making predictions about its increasing or decreasing behavior, as well as identifying important points such as inflection points.
A function is convex if its graph curves upward, while a function is concave if it curves downward. This can also be determined by looking at the second derivative of the function. If the second derivative is positive, the function is convex, and if it is negative, the function is concave.
Inflection points are points on a function where the concavity changes. This means that the function goes from being convex to concave, or vice versa. Inflection points are important because they can help identify key points of change in a function, and can also be used to determine the behavior of a function around those points.
To find these intervals, you can first find the critical points of the function by setting the first derivative equal to zero. Then, use the second derivative test to determine the concavity at these points. The intervals between these critical points will correspond to intervals where the function is either convex or concave.
Knowing these intervals can help in understanding the overall shape and behavior of a function. It can also be useful in making predictions about the behavior of the function, such as identifying where it is increasing or decreasing, and finding important points such as local maxima or minima.