Sequences & Series
AM-GM Inequality — Maximum of Product
nta_pyq_2023_apr
Grade 11
Question:
Let $a,b,c,d>0$ with $a+b+c+d=11$. If max of $a^5b^3c^2d=3750\beta$, then $\beta$ is
Step-by-Step Solution
Key Concept: By weighted AM-GM: $\frac{5(a/5)+3(b/3)+2(c/2)+d}{11}\geq(a^5b^3c^2d)^{1/11}\cdot\frac{1}{5^5\cdot3^3\cdot2^2}^{1/11}$... use AM-GM with $a$ split 5 ways, $b$ 3 ways, etc.
$\beta=90$.
Correct Answer: 1