TS EAMCET · Maths · Application of Derivatives
\(x\) and \(y\) are two positive integers such that \(2 x+3 y=50\). If \(x^2 y^3\) is maximum for \(x=\alpha\) and \(y=\beta\), then \(\frac{\alpha}{2}+\frac{\beta}{5}=\)
- A \(10\)
- B \(\frac{10}{3}\)
- C \(5\)
- D \(7\)
Answer & Solution
Correct Answer
(D) \(7\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text 2 x+3 y=50 \\ & P=x^2 y^3 \text { (given) } \\ & \Rightarrow P=x^2\left(\frac{50-2 x}{3}\right)^3 \Rightarrow P=\frac{1}{27} x^2(50-2 x)^3 \\ & \Rightarrow \frac{d P}{d x}=\frac{1}{27}\left[x^2 \cdot 3(50-2 x)^2(-2)+2 x \cdot(50-2 x)^3\right] \\ & \text {…
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