Polynomial/graphing question

Question

Given y = x^3 + 10x^2 + 4x + 1, match the points with the appropriate descriptions

(-8, 97)

(2, 57)

(-10/3, 1667/27)

A: the function is increasing and convex at this point

B: the function is increasing and concave at this point

C: this is an inflection point

Answer 1

Given y = x³+10x²+4x=1

dy/dx = 3x²+20x+4

d²y/dx² = 6x +20

d²y/dx² < 0 at (-8, 97) so it is concave and increasing

d²y/dx² > 0 at (2, 57) so it is convex and increasing

For point of inflection, d²y/dy² = 0

so substitute (-10/3) into d²y/dy² = 6(-10/3) + 20 = 0 so it is point of inflection