Answer :
Final answer:
The intervals of concavity and points of inflection of a function are found using the second derivative. Points of inflection occur where the second derivative equals zero. The sign of the second derivative defines the intervals of concavity for the function.
Explanation:
To find the intervals of convexity and points of inflections for the function
y = x⁴-12x³+48x²-50
, we first need to find the second derivative of the function. Firstly, find the first derivative y' = 4x³-36x²+96x. Then, find the second derivative y'' = 12x²-72x+96. The intervals of convexity are determined by where y'' is positive (convex up) or negative (convex down). Set y'' = 0 to find the potential points of inflection, i.e., 12x²-72x+96 = 0. Solve this equation to get the x-values of possible points of inflection. Substitute these x-values into the original equation to get the corresponding y-values. These x,y pairs are your points of inflection. The intervals of convexity are the ranges of x where the function is concave up or down based on the sign of y''.
Learn more about Intervals of Convexity here:
https://brainly.com/question/30463293
#SPJ11
