Find two numbers whose difference is 150 and whose product is a minimum.

Question

misty1212 · Answer

HI!!

anonymous · Answer

hi!

misty1212 · Answer

lets call one of the numbers $x$
then the other must be $150+x$ since there difference is $150+x-x=150$

misty1212 · Answer

the product is therefore 
$$P(x)=x(150+x)$$ and you want to minimize that one

misty1212 · Answer

$$P(x)=150x+x^2$$ is a parabola that opens up
the minimum is at the vertex 
the first coordinate of the vertex of $y=ax^2+bx+c$ is $-\frac{b}{2a}$ which in your example is 
$$\frac{150}{2}=-75$$

misty1212 · Answer

that makes the other one $150-75=75$ and those are your two numbers

anonymous · Answer

So you don't need to take the derivative of P(x)=150x+x^2?

misty1212 · Answer

lol no not unless you are a slave to calculus
no one needs calculus to find the vertex of a parabola
but you will get the same answer if you do it that way

misty1212 · Answer

the derivative is $150-2x$ set it equal to zero and you still get $-75$

misty1212 · Answer

oops i meant the derivative is $150+2x$ doe

anonymous · Answer

Okay, Thank you! :) @misty1212

misty1212 · Answer

$$\color\magenta\heartsuit$$