Finding Extrema for (x-1)x^(2/3)

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Homework Statement



i have to find the extrema for (x-1)x^(2/3)
so i found the derivative: (5x/3)^2/3 - 2/(3x^1/3)
now i have to find the critical numbers by putting the derivative equal to zero and solving for x...but i don't know how to do it.
are there even any roots?
 
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So you have
\left(\frac{5x}{3}\right)^{2/3} - \frac{2}{3x^{1/3}}= 0[/itex]<br /> If x= 0 then the last fraction is undefined (so x= 0 <b>is</b> a critical point), if not we can multiply through by 3x^{1/3}. What does that give you?
 
if you plug in zero though, it becomes undefined, not 0...so how can that be a critical point? anddo you mean taking (5x-2)/(3x^1/3)
and since the bottom doesn't matter, jst find the zero of the top part (5x-2) and get 2/5?
would that be correct?
and the only root?
 
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menal said:
if you plug in zero though, it becomes undefined, not 0...so how can that be a critical point? and do you mean taking (5x-2)/(3x^1/3)
and since the bottom doesn't matter, just find the zero of the top part (5x-2) and get 2/5?
would that be correct?
and the only root?
Review the definition of a critical number when finding extrema.
 
oh! lol sorry. i ever knew that! so 0 is a critical number along with 2/5. that would be it?
 
menal said:
oh! lol sorry. i ever knew that! so 0 is a critical number along with 2/5. that would be it?
How did you get 2/5?
 
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"you mean taking (5x-2)/(3x^1/3)
and since the bottom doesn't matter, just find the zero of the top part (5x-2) and get 2/5"
 
menal said:
"you mean taking (5x-2)/(3x^1/3)
and since the bottom doesn't matter, just find the zero of the top part (5x-2) and get 2/5"
Right. Now, use the critical numbers to find the extrema.
 
yup! the interval given was (0,1)
so i plugged in 0,1 and 2/5 into the first equation:(x-1)x^(2/3)
and got 0,0 and -0.32
so my extreme values would be -0.32 as the absolute minimum and 0 as the absolute max.
 
  • #10
Was the interval (0,1), or was it [0,1] ?

Is x = 0 in your interval?
 
  • #11
[0,1]
 
  • #12
Look at both ends of the interval.
 
  • #13
you mean -2/5 would be one too?
 
  • #14
Of course. But when looking for extrema on a closed interval, you need to check each end of the interval for a possible extremum.
 
  • #15
so your sayin that -2/5 and 2/5 and then 7/5 (1 + 2/5) and 3/5 (1 - 2/5) would all be critical numbers.
 
  • #16
No.

If you're looking for extrema on the interval [0,1], the end points of the interval are also critical numbers. The end points are, x=0 (already a critical number) and x=1.

-2/5 is not in the interval [0,1] and not a critical number.
 
  • #17
yes, i have included 0 and 1 and 2/5 as my critical numbers already. I was just confused because i thought u were saying that i had to add/subtract the crticial number i got (2/5) from my intervals (0 and 1). But no, i think i get it now. the crticial numbers will be 0,1 and 2/5.
 
  • #18
Yes. 0, 2/5, and 1 .
 
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