Q97 / 200 · AP EAMCET 2001 · Mathematics

The maximum value of $x y$ subject to $x+y=7$ is

  • A
    $10$
  • B
    $12$
  • Correct answer
    C
    $\frac{49}{4}$
  • D
    $\frac{55}{4}$

Solution

Let $M=x y$ $\begin{aligned} M & =x(7-x)=7 x-x^2 \quad[\because x+y=7 \text { given }] \\ \frac{d M}{d x} & =7-2 x\end{aligned}$ For maximum or minimum, put $\frac{d M}{d x}=0$ $x=\frac{7}{2} \Rightarrow \frac{d^3 M}{d x^2}=-2=-v e$ The maximum value at $x=\frac{7}{2}$ $=\frac{7}{2}\left(7-\frac{7}{2}\right)=\frac{7}{2}\left(\frac{7}{2}\right)=\frac{49}{4}$

Hence, the correct answer is option: 'C'.

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