Calc Brain Fart HELP thbbb thbbb thbbb

Loren Michael

What I have so far:

C(x) = 8(x^2 - 8x + 20)^(1/2) + 4x

C' = 8(1/2)(2x-8)(x^2 - 8x + 20)^(-1/2) + 4

Now, this is all in the book that I'm working with, which is riddled with errors, but apparently I can go from that to this:

3x^2 - 24x + 44 = 0

And I'm just not getting how it gets from step two to step three. And the book has betrayed me enough times that I'm not sure if it's my own (bad. terrible. really really awful.) intuition being right this time, or if it's the book being wrong, which it is fairly often.

Help!

Dance Commander

Well, without setting either C(x) = 0 or C'(x) = 0, there's no way to even get close. What are you trying to do?

TechBoy

He's solving for critical points, so setting C'(x) = 0.

If you do that and just run through the algebra you get the book's answer.

Gaffero

C' = (8x-32)(x^2 - 8x + 20)^(-1/2) + 4

I'm not sure how to get from step 2 to step 3, but I'd try rewriting your first derivative so that the (x^2 -8x+20) goes in the denominator with 8x-32 in the numerator, the whole thing plus 4:

..........8x-32.................4

---

+ ----
sqrt ((x^2 -8x+20).......1

Then simplify under a single denominator:

(8x-32) +4sqrt(x^2 -8x +20)

---

sqrt (x^2 -8x+20)

From here I'm less sure of how to get the square roots out; perhaps an algebra-savy PAer can pick it up from here.

What I do know that the function attains a minimum at x=2.85. I confirmed this by graphing and checking the given answer in your book. You'll need to work the algebra magic to get to Step 3 so you can use the quadratic formula to solve and get:

[24 +/- sqrt(48)]/(6)
or in words
24 plus or minus the square root of 48 all over six.

[24 - sqrt(48)]/6 is roughly 2.85; your minimum is (2.85, 29.85)

BlochWave

Just back up to this step

0 = (8x-32)(x^2 - 8x + 20)^(-1/2) + 4

subtract 4 from both sides

-4 = (8x-32)(x^2-8x+20)^(-1/2)

square both sides (the -1/2 becomes -1)

16=(8x-32)^2 / (x^2-8x+20), multiply by the right hand denominator

16*x^2-128x+320=(8x-32)^2, multiply out the right hand, (8x-32)(8x-32) = 64x^2 -256x - 256x + 1024

64x^2-512x+1024 for the right hand

16*x^2-128x+320=64x^2-512x+1024 ahhhh look at all the RAM sizes

simplify (I moved the left hand stuff to the right hand side)

48x^2-384x+704 = 0 Factor out 4

4(12x^2-96x + 176)=0 divide by 4

12x^2-96x+176=0 (0/4=0)

oo we can do 4 again (we could've factored 16 the first time but I didn't see it)

3x^2-24x+44 = 0 ,because again we divide out the factored out term

QED etc.

Your intuition that your intuition is bad is good Hardest part of calculus is the algebra. Hardest part of DEs is the calculus. etc.