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Use the methods of this section to sketch the curve $ y = x^3 - 3a^2 x + 2a^3 $, where $ a $ is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?

The graph above is for a > 0For $a=0,$ we have a special case since the function will be just $y=x^{3}$ andthus will have no local minima or maxima.For $a < 0,$ the graph will be shifted 4$a^{3}$ units down

03:50

Wen Z.

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 3

How Derivatives Affect the Shape of a Graph

Derivatives

Differentiation

Volume

Missouri State University

Harvey Mudd College

University of Nottingham

Boston College

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I say we sketched the graph like this. As you can see, if a is greater than zero, we can see the graph. And then for a equal zero, we conceded we would just have a cube function. Why equals X Cube? You know there's no local minimums or Maxima for a cubic function. Therefore, we know that when a is less than zero, the graphics shifted for a cube units down. So this is down.

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