Find two numbers differing by 46 whose product is as small as possible.

Answer:The numbers are 23 and -23.Step-by-step explanation:If the two numbers are x and y, we can say that the product is[tex]P=x\cdot y[/tex]We need to eliminate a variable. To do that we use the fact that the difference between the two numbers has to be 46.[tex]x-y=46[/tex]So we can say [tex]y=x-46[/tex] and plug that into the equation for P[tex]P=x\cdot (x -46)=x^2-46x[/tex]From the information given we want the product P to be minimized. For this we find the derivative[tex]\frac{d}{dx}(x^2-46x) =\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}\left(46x\right)=2x-46[/tex]Next, we find the critical points, we set the above equation equal to zero and solve for x[tex]2x-46=0\\2x-46+46=0+46\\2x=46\\\frac{2x}{2}=\frac{46}{2}\\x=23[/tex]Since x = 23 is the only critical number, we can conclude that there’s actually an absolute minimum at x = 23Thus, [tex]x = 23[/tex] and [tex]y = 23-46=-23[/tex]