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Problem 58 Medium Difficulty

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?

Answer

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

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Video Transcript

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.