$9a^3 y = x(4a - x)^3$
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.
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 I think something is wrong by Jhun Vert
The same error. You can
In reply to Ay sorry po. 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 The same error. You can by Jhun Vert
i-check ko. Anong page?
In reply to Kinuha ko kasi sa libro ni by Francis June E…
i-check ko. Anong page?
62-64 sir
In reply to i-check ko. Anong page? by Jhun Vert
Okay I got it. The given is
In reply to i-check ko. Anong page? by Jhun Vert
May answer key po ba kayo
In reply to i-check ko. Anong page? by Jhun Vert
.
$9a^3 y = x(4a - x)^3$
Maraming Salamat po sir.
In reply to $9a^3 y = x(4a - x)^3$ by Jhun Vert
Sir? Paano po naging (4a-4x)
Okay na po pala sir hehe.
May answer key po ba kayo
1
y=x³-3x+1 Soln:
y=x³-3x+1
Soln: