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
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I think something is wrong

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.

Ay sorry po.

In reply to by Jhun Vert

Ay sorry po. Eto po sir 9a²y=x(4a-x)³

The same error. You can

In reply to by Francis June E…

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

Kinuha ko kasi sa libro ni

In reply to by Jhun Vert

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

i-check ko. Anong page?

In reply to by Francis June E…

i-check ko. Anong page?

Okay I got it. The given is

In reply to by Jhun Vert

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

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

$9a^3 y = x(4a - x)^3$   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

Maraming Salamat po sir.

In reply to by Jhun Vert

Maraming Salamat po sir. Godbless you po. :)

Sir? Paano po naging (4a-4x)

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

Okay na po pala sir hehe.

Okay na po pala sir hehe. Salamat po

1

y=x³-3x+1
Soln: