# Chapter 4 - Section 4.3 - How Derivatives Affect the Shape of a Graph - 4.3 Exercises - Page 302: 58

All the curves in this family have the following in common: The local maximum is $(-a, 4a^3)$ The local minimum is $(a, 0)$ $y$ is concave down on the interval $(-\infty, 0)$ $y$ is concave up on the interval $(0, \infty)$ The point of inflection is $(0, 2a^3)$ The general shape of the graph is similar for all the curves in this family. However, the specific points of local maximum, local minimum, intercepts, and point of inflection depend on the value of $a$. $y = x^3-3a^2x+2a^3$ We can find the points where $y' = 0$: $y' = 3x^2-3a^2 = 0$ $3x^2 = 3a^2$ $x = \pm a$ When $x=-a$, then $y = (-a)^3-3a^2(-a)+2a^3 = 4a^3$ When $x=a$, then $y = (a)^3-3a^2(a)+2a^3 = 0$ The local maximum is $(-a, 4a^3)$ The local minimum is $(a, 0)$ We can find the points where $y'' = 0$: $y'' = 6x = 0$ $x = 0$ $y$ is concave down on the interval $(-\infty, 0)$ $y$ is concave up on the interval $(0, \infty)$ The point of inflection is $(0, 2a^3)$ The general shape of the graph is similar for all the curves in this family. However, the specific points of local maximum, local minimum, intercepts, and point of inflection depend on the specific value of $a$. 