Question

Find the two numbers whose sum is 15 and whose product is maximum.

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{Sum of two numbers is 15}$

$\underline{\textbf{To find:}}$

$\textsf{The two numbers whose product is maximum}$

$\underline{\textbf{Solution:}}$

$\textsf{Let the two numbers x and y (x > y)}$

$\textsf{As per given data,}$

$\mathsf{x+y=15}$

$\implies\mathsf{y=15-x}$

$\textsf{Their product is xy}$

$\mathsf{P=xy}$

$\mathsf{P=x(15-x)}$

$\mathsf{P=15x-x^2}$

$\mathsf{P(x)=15x-x^2\;(say)}$

$\mathsf{P'(x)=15-2x}$

$\mathsf{P''(x)=-2}$

$\mathsf{For\;maximum,\;P'(x)=0}$

$\implies\mathsf{15-2x=0}$

$\implies\mathsf{2x=15}$

$\implies\mathsf{x=\dfrac{15}{2}}$

$\implies\boxed{\mathsf{x=7.5}}$

$\mathsf{when\;x=7.5,\;P''(x)=-2\;<\;0}$

$\therefore\textsf{P(x) is maximum when x=7.5}$

$\mathsf{x=7.5\;\implies\;y=7.5}$

$\textsf{Hence the required numbers are 7.5 and 7.5}$