Let a , b , c and d be positive real numbers such that a + b + c + d = 11 . If the maximum value of a 5 b 3 c 2 d is 3750, then the value of is [2023]

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Solution

Q.
Let $a, b, c$ and $d$ be positive real numbers such that $a + b + c + d = 11 .$ If the maximum value of $a^{5} b^{3} c^{2} d$ is $3750 β$ , then the value of $β$ is [2023]

1 55

2 108

3 90

4 110

Ans.
(3)

$Given: a + b + c + d = 11$

$\frac{5 (\frac{a}{5}) + 3 (\frac{b}{3}) + 2 (\frac{c}{2}) + d}{11} \geq {(\frac{a^{5} b^{3} c^{2} d}{5^{5} 3^{3} 2^{2}})}^{\frac{1}{11}}$

$\Rightarrow 1 \geq {(\frac{a^{5} b^{3} c^{2} d}{5^{5} 3^{3} 2^{2}})}^{\frac{1}{11}} \Rightarrow a^{5} b^{3} c^{2} d \leq 5^{5} 3^{3} 2^{2}$

$∴$ $Max of a^{5} b^{3} c^{2} d = 5^{5} 3^{3} 2^{2}$

$\Rightarrow 3750 β = 5^{5} 3^{3} 2^{2} \Rightarrow β = 90$

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