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Let $2x+3y+4z=9$, x, y, z > $0$ then the maximum value of ${{(1+x)}^{2}}{{(2+y)}^{3}}{{(4+z)}^{4}}$ is
A) ${{2}^{8}}$
B) ${{4}^{9}}$
C) ${{\left( \dfrac{11}{3} \right)}^{9}}$
D) ${{\left( \dfrac{11}{6} \right)}^{9}}$

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Last updated date: 16th May 2024
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Answer
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Hint: The maximum value of any function is denoted at the values of variables x, y and z which makes the value of the function the highest value among all possible values of x, y and z.
The values of function to the left of maximum are rising while at the right, its values are falling.

Complete step by step solution:
It is given that a = $2x+3y+4z=9$. The maximum value of function, b = ${{(1+x)}^{2}}{{(2+y)}^{3}}{{(4+z)}^{4}}$ is to be determined.
As functions a and b are related in terms of powers and bases, it can be said that
$\Delta b=\lambda \Delta a$
$\lambda $ is an arbitrary value.
Δb is determined as:
$
 b={{(1+x)}^{2}}{{(2+y)}^{3}}{{(4+z)}^{4}} \\
\Delta b=\dfrac{db}{dx},\dfrac{db}{dy},\dfrac{db}{dz} \\
\Delta b = 2(1+x){{(2+y)}^{3}}{{(4+z)}^{4}},3{{(1+x)}^{2}}{{(2+y)}^{2}}{{(4+z)}^{4}},4{{(1+x)}^{3}}{{(2+y)}^{3}}{{(4+z)}^{3}}
$
Δa is determined as:
$
 a=2x+3y+4z \\
 \Delta a=\dfrac{da}{dx},\dfrac{da}{dy},\dfrac{da}{dz} \\
 \Delta a=2,3,4
$
By putting values of Δb and Δa in above relation,
$\begin{align}
  & \Delta b=\lambda \Delta a \\
 & \left[ 2(1+x){{(2+y)}^{3}}{{(4+z)}^{4}},3{{(1+x)}^{2}}{{(2+y)}^{2}}{{(4+z)}^{4}},4{{(1+x)}^{3}}{{(2+y)}^{3}}{{(4+z)}^{3}} \right]=\lambda [2,3,4] \\
\end{align}$
On comparing,
$\begin{align}
  & \left[ 2(1+x){{(2+y)}^{3}}{{(4+z)}^{4}} \right]=\lambda \times 2 \\
 & (1+x){{(2+y)}^{3}}{{(4+z)}^{4}}=\lambda \text{ -eq(1)}
\end{align}$
$\begin{align}
  & 3{{(1+x)}^{2}}{{(2+y)}^{2}}{{(4+z)}^{4}}=\lambda \times 3 \\
 & {{(1+x)}^{2}}{{(2+y)}^{2}}{{(4+z)}^{4}}=\lambda \text{ -eq(2)}
\end{align}$
$\begin{align}
  & 4{{(1+x)}^{3}}{{(2+y)}^{3}}{{(4+z)}^{3}}=\lambda \times 4 \\
 & {{(1+x)}^{3}}{{(2+y)}^{3}}{{(4+z)}^{3}}=\lambda \text{ -eq(3)}
\end{align}$
Solving three equations gives,
x = \[\dfrac{8}{3}\] , y = \[\dfrac{5}{3}\], z = \[\dfrac{-1}{3}\].
Maximum value of function b is calculated by putting the determined values of x, y and z at which function b is maximized.
Maximized value of function b = ${{(1+x)}^{2}}{{(2+y)}^{3}}{{(4+z)}^{4}}$ at x = \[\dfrac{8}{3}\] , y = \[\dfrac{5}{3}\], z = \[\dfrac{-1}{3}\]
Maximized value = ${{\left( 1+\dfrac{8}{3} \right)}^{2}}{{\left( 2+\dfrac{5}{3} \right)}^{3}}{{\left( 4+\dfrac{\left( -1 \right)}{3} \right)}^{4}}$ = ${{\left( \dfrac{11}{3} \right)}^{2}}{{\left( \dfrac{11}{3} \right)}^{3}}{{\left( \dfrac{11}{3} \right)}^{4}}={{\left( \dfrac{11}{3} \right)}^{2+3+4}}={{\left( \dfrac{11}{3} \right)}^{9}}$
This indicates that option $(3)$ is correct.

Note:
Maximum value of function is calculated by taking a derivative so as to find the number below which the value of function is rising and after which the value of function is falling.
Maximized function has a peak as is the peak of a mountain. Firstly, it rises, reaches the maximum and then falls.