a+b=20, find the maximum value of a²b
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Solution
Finding the maximum value of an expression like \(a^2 b\) under a given constraint such as \(a + b = 20\) is a classic optimization problem in algebra. It beautifully illustrates the balance between variables when one quantity increases and the other decreases.
In this post, we’ll determine step-by-step how to maximize \(a^2 b\) using substitution and differentiation, and we’ll uncover the value of \(a\) and \(b\) that makes this expression reach its peak.
This kind of problem is not only common in mathematics competitions but also serves as an elegant application of calculus and reasoning about proportional relationships.
\[
\begin{aligned}
a+b&=20 \\\\
b&=20-a \\\\
a^2 b&=a^2(20-a) \\\\
&=20 a^2-a^3\\\\
f(a) & =-a^3+20 a^2 \\\\
f^{\prime}(a) & =-3 a^2+40 a \\\\
& =a(-3 a+40)\\\\
f^{\prime}(a)&=0 \\\\
a&=0\\\\
\text {or}\\\\
a&=\frac{40}{3} \simeq 13,33\\\\
& a=13 \text { and } b=7 \Rightarrow a^2 b=13^2 \cdot 7=1183 \\\\
& a=14 \text { and } b=6 \Rightarrow a^2 b=14^2 \cdot 6=1176\\\\
& \text { Thus, The maximum value of}\text { } a^2 b \text { is } 1183
\end{aligned}
\]
Home -> Solved problems -> a+b=20, find the maximum value of a²b
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