Graphing The function y = x^(4/3)-4x^(1/3)

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juanjjuarez0

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Help me to solve this question using first and second derivative. :)
I did this but something went wrong I guess:
-->Calculated critical and inflection point using 1st and second derivative.
-->Tested the increasing and decreasing nature at all possible points using first derivative.
--> Calculated maximum and minimum values at critical points but amazingly both were 0. :(
--> Tested the concavity of the function using second derivative.
The graph was somehow hilarious! Sorry I couldn't attach the work sheet here. Anyway help me out please.
 
The steps you've outlined sound to me like a perfectly fine way to start. I'll add a few to the list of things you might consider though:

  • What are the roots of the function?
  • Are there any y-intercepts of the function? If there are, what are they?
  • What is the behavior as the function grows unbounded?
  • What is the behavior as the function grows unbounded in the negative direction?

When it comes to making graphs by hand, they're pretty much always going to look a bit off and be somewhat inaccurate. Did you "cheat" and graph it using a calculator or online tool, such as Desmos? Does your proposed solution at least look close? If your graph is way off in left field, please share all of your work with us, even the stuff you know is wrong. In particular, what did you get as the first and second derivatives? What were the critical and inflection points those derivatives gave you?
 
pka said:
I would first factor then graph.
\(\displaystyle y=x^{4/3}-4x^{1/3}=\sqrt[3]{x}(x-4)\).

See the plot. \(\displaystyle \ \ \ \ \ \)This doesn't plot the desired graph. However, **
Click to expand...

\(\displaystyle y = x^{\tfrac{1}{3}} \ \ \ and \ \ \ y = \sqrt[3]{x} \ \) are equivalent, but WolframAlpha doesn't handle them the same.

A way around this is to type "plot cuberoot(x)" (without the quotation marks and in some similar shorthand
of characters) in WolframAlpha to see that correct graph, if that were the exercise.

For the problem at hand, you may type "plot cuberoot(x)*(x - 4)" (without the quotation marks and in some
similar shorthand of characters) in WolframAlpha to see the correct graph.



**At www.quickmath.com under "Equations," "Plot," click "Advanced," click "Grid Lines," and enter
y = x^(4/3) - 4^(1/3) or enter y = x^(1/3)*(x - 4) to see the correct graph also.
 
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