a+b=20, find the maximum value of a²b

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\[ \begin{aligned} a+b&=20 \\\\ b&=20-a \\\\ a^2 b&=a^2(20-a) \\\\ &=20 a^2-a^3\\\\ f(a) & =-a^3+20 a^2 \\\\ f^{\prime}(a) & =-3 a^2+40 a \\\\ & =a(-3 a+40)\\\\ f^{\prime}(a)&=0 \\\\ a&=0\\\\ \text {or}\\\\ a&=\frac{40}{3} \simeq 13,33\\\\ & a=13 \text { and } b=7 \Rightarrow a^2 b=13^2 \cdot 7=1183 \\\\ & a=14 \text { and } b=6 \Rightarrow a^2 b=14^2 \cdot 6=1176\\\\ & \text { Thus, The maximum value of}\text { } a^2 b \text { is } 1183 \end{aligned} \]

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Home -> Solved problems -> a+b=20, find the maximum value of a²b

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