Minima Maxima: 9a³y=x(4a-x)³

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Francis June E....
Francis June E. Seraspe's picture
Minima Maxima: 9a³y=x(4a-x)³

Patulong po ulit ako mam/sir
Yung process po sana ulit
9a²y=x(4a+x)³
Yung ans nya po is (a,3a) maximum
Thanks in advance mam/sir

Jhun Vert
Jhun Vert's picture

I think something is wrong with your given. If you check for x = a, y ≠ 3a. Here is the checking of your answer key.

Set x = a and solve for y
$9a^2 y = a(4a + a)^3$

$9a^2 y = 125a^4$

$y = \dfrac{125a^2}{9}$   ←   not equal to 3a
 

Kindly check the given.
 

Francis June E....
Francis June E. Seraspe's picture

Ay sorry po.
Eto po sir 9a²y=x(4a-x)³

Jhun Vert
Jhun Vert's picture

The same error. You can evaluate if x = a, y = 3a2 not 3a. Either the given is wrong or the answer key is wrong.

Francis June E....
Francis June E. Seraspe's picture

Kinuha ko kasi sa libro ni Love and Rainville yang example sir.

Jhun Vert
Jhun Vert's picture

i-check ko. Anong page?

Francis June E....
Francis June E. Seraspe's picture

62-64 sir

Jhun Vert
Jhun Vert's picture

Okay I got it. The given is actually $9a^3 y = x(4a - x)^3$

Jhun Vert
Jhun Vert's picture

$9a^3 y = x(4a - x)^3$
 

Differentiate y in terms of x using product formula
$9a^3 y' = -3x(4a - x)^2 + (4a - x)^3$
 

Simplify (optional)
$9a^3 y' = (4a - x)^2 [-3x + (4a - x)]$

$9a^3 y' = (4a - x)^2 (4a - 4x)$

$9a^3 y' = 4(4a - x)^2 (a - x)$
 

Determine the 2nd derivative
$9a^3 y'' = -4(4a - x)^2 - 8(4a - x)(a - x)$

$9a^3 y'' = -4(4a - x)[ (4a - x) + 2(a - x) ]$

$9a^3 y'' = -4(4a - x)(6a - 3x)$

$9a^3 y'' = -12(4a - x)(2a - x)$
 

Set y' to zero for maxima and/or minima
$0 = 4(4a - x)^2 (a - x)$
 

For (4a - x)2 = 0:

$x = 4a$
 

$9a^3 y = 4a(4a - 4a)^3$

$y = 0$
 

Check if Maxima or Minima.
$9a^3 y'' = -12(4a - 4a)(2a - 4a)$

$y'' = 0$   ←   inflection point (the curve is neither upward nor downward)
 

Hence, (4a, 0) is neither maximum nor minimum

 

For a - x = 0

$x = a$
 

$9a^3 y = a(4a - a)^3$

$y = 3a$
 

Check if Maxima or Minima.
$9a^3 y'' = -12(4a - a)(2a - a)$

$y'' = (-)$   ←   the curve is concave downward
 

Hence, (a, 0) is maximum.

 

Here is the graph of the curve with a = 1
 

screenshot_2020-04-10_14.23.04.png

 

Francis June E....
Francis June E. Seraspe's picture

Maraming Salamat po sir. Godbless you po. :)

Francis June E....
Francis June E. Seraspe's picture

Sir? Paano po naging (4a-4x) ? Yung sa simplify po

Francis June E....
Francis June E. Seraspe's picture

Okay na po pala sir hehe. Salamat po

 
 
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