a function has a local maximum at x=-2 and x=6 and a local minimum at x=1. how do u find the concavity of this function and point of inflection??
To find the concavity just graph it or use the first derivative and pick values greater than, or less than your critical points see if those values are positive or negative. IF positive, the graph will concave up, if negative - concave downwards.
To find the points of inflection, find the second derivative and set equal to zero. Once that has been done, plug that X value into the original and it will output a Y value. And that point (X,Y) will be your critical point.
but how do i find point of inflection from the given local max and mins
To find the concavity and point of inflection of a function, you need to analyze the second derivative of the function.
Step 1: Find the first derivative of the function.
Differentiate the function once to find its first derivative. The critical points of the first derivative will help identify the local maximum and minimum.
Step 2: Find the second derivative of the function.
Differentiate the first derivative (obtained in Step 1) to find the second derivative. The second derivative provides information about the concavity and points of inflection.
Step 3: Analyze the second derivative to identify the concavity.
a) Test the sign of the second derivative at each critical point found in Step 1. If the second derivative is positive, the function is concave up in that interval. If the second derivative is negative, the function is concave down.
b) If the sign of the second derivative changes from positive to negative or from negative to positive at a critical point, that point is a point of inflection.
So here is what you should do:
Step 1: Find the first derivative.
Differentiate the function to find its first derivative.
Step 2: Find the second derivative.
Differentiate the first derivative obtained in Step 1 to find the second derivative.
Step 3: Analyze the second derivative to identify the concavity and points of inflection.
Identify the critical points from the first derivative. Test the sign of the second derivative at these points to determine concavity. Look for sign changes in the second derivative to identify the points of inflection.
By following these steps, you can find the concavity and points of inflection of the given function.