Concavity calculus

 
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ケネス ケネス's picture
Concavity calculus

Here's my question:
Example: examine the function
Y=x(x-1)^3
Y'=3x(x-1)^2+(x-1)^3=(x-1^2(4x-1)
Y"=4(x-1)^2+2(x-1)(4x-1) ... How did they get 4(x-1)^2+2(x-1)(4x-1)?

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Jhun Vert's picture
$y = x(x - 1)^3$

$y = x(x - 1)^3$

$y' = 3x(x - 1)^2 + (x - 1)^3$

$y' = (x - 1)^2 [ \, 3x + (x - 1) \, ]$

$y' = (x - 1)^2 (4x - 1)$
 

Use $d(uv) = u \, dv + v \, du$

$u = (x - 1)^2$     →     $du = 2(x - 1)$

$v = 4x - 1$   →     $dv = 4$

 
$y'' = (x - 1)^2 (4) + (4x - 1)[ \, 2(x - 1) \, ]$

$y'' = 4(x - 1)^2 + 2(4x - 1)(x - 1)$   ←   this is your question
 

You can also go further by factoring...
$y'' = 2(x - 1) [ \, 2(x - 1) + (4x - 1) \, ]$

$y'' = 2(x - 1)(6x - 3)$

ケネス ケネス's picture
Lastly, the next equation

Lastly, the next equation says that 6(x-1)(2x-1). How did they get 6(x-1)(2x-1)?

Alexander's picture
From y'' = 2(x − 1)(6x − 3),

From y'' = 2(x − 1)(6x − 3), factor 3 from (6x - 3) and it will become y'' = 2(x − 1)(3)(2x − 1).

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