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# Find two numbers whose sum is $100$ and whose product is a maximum.

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## 1 Answer

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Toolbox:
• Second derivative test: Suppose $f$ is continuous on an open interval that contains
• (i)If $f'(c)=0\; and\; f''(c) >0$ then $f$ has a local minimum at $c$.
• (ii)If $f'(c)=0\;and \; f''(c)<0$ then $f$ has a local maximum at $c$
Step 1:
Let the two numbers be x and 100 - x (since their sum is 100)
Step 2:
Their product is $p= x(100-x)$
$\qquad=100x -x^2$
Step 3:
$\large\frac{dp}{dx}$$=100-2x At extreme values of P, \large\frac{dp}{dx}$$=0 =>100-2x=0$
$\qquad=>x=50$

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