http://25.media.tumblr.com/tumblr_mdebc6…
I got stuck after finding the first derivative... :/
I got stuck after finding the first derivative... :/
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That's an ugly derivative!
f = (x-6)^1/3(2x-6)^2/3
f' = 1/3(x-6)^-2/3(2x-6)^2/3 + 2/3(2x-6)^-1/3(2)(x-6)^1/3
= [(2x-6)^2/3]/[3(x-6)^2/3] + [4(x-6)^1/3]/[3(2x-6)^1/3]
= [2x-10]/[(x-6)^1/3(2x-6)^1/3
critical numbers are where f'=0 or is undefined (i.e. denominator=0)
2x-10=0 ---> x=5
x-6=0 ---> x=6
2x-6=0 ----> x=3
so critical numbers at x=3, 5, 6
To do first derivative test, choose values on either side of these critical numbers and plug into the derivative equation to see if the slope is positive or negative.
f'(0) = 1.667 positive slope
f'(4) = -1 negative slope
f'(5.5) = .92832 positive slope
f'(7) = 2 positive slope
The function is decreasing on (3,5) and increasing everywhere else
f = (x-6)^1/3(2x-6)^2/3
f' = 1/3(x-6)^-2/3(2x-6)^2/3 + 2/3(2x-6)^-1/3(2)(x-6)^1/3
= [(2x-6)^2/3]/[3(x-6)^2/3] + [4(x-6)^1/3]/[3(2x-6)^1/3]
= [2x-10]/[(x-6)^1/3(2x-6)^1/3
critical numbers are where f'=0 or is undefined (i.e. denominator=0)
2x-10=0 ---> x=5
x-6=0 ---> x=6
2x-6=0 ----> x=3
so critical numbers at x=3, 5, 6
To do first derivative test, choose values on either side of these critical numbers and plug into the derivative equation to see if the slope is positive or negative.
f'(0) = 1.667 positive slope
f'(4) = -1 negative slope
f'(5.5) = .92832 positive slope
f'(7) = 2 positive slope
The function is decreasing on (3,5) and increasing everywhere else