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Question

(A) Let the two numbers be $x$ and $y$.Given that their sum is fixed,let $x + y = s$,where $s$ is a constant.Then $y = s - x$.Let the product be $f(x)

Vedclass · Accepted Answer

Each number is half of the sum

Vedclass · Answer

(A) Let the two numbers be $x$ and $y$.Given that their sum is fixed,let $x + y = s$,where $s$ is a constant.Then $y = s - x$.Let the product be $f(x) = xy = x(s - x) = sx - x^2$.To find the maximum product,we differentiate $f(x)$ with respect to $x$:$f'(x) = s - 2x$.Setting $f'(x) = 0$ for critical points:$s - 2x = 0 \Rightarrow x = s/2$.Since $f''(x) = -2 Substituting $x = s/2$ into $y = s - x$,we get $y = s - s/2 = s/2$.Thus,the product is maximum when each number is half of the sum.

Column-$I$	Column-$II$
$(A)$ If $(a, b)$ is a point of inflection,then $a - b$ equals	$(P)$ $3$
$(B)$ If $e^t$ is the point of local minimum,then $12t$ equals	$(Q)$ $1$
$(C)$ If the graph is concave down on $(d, e)$,then $d + 3e$ equals	$(R)$ $4$
$(D)$ If the graph is concave up on $(p, \infty)$,then $p$ equals	$(S)$ $8$

The sum of two numbers is fixed. Then their product is maximum when:

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