Question

find two non negative numbers x and y whose sum is 100 and for which x^2 y is a maximum

Answer 1

did you look on google

Answer 2

yea

Answer 3

Hey there,
\(\Large\bf \color{#CC0033}{\text{Welcome to OpenStudy! :)}}\)

Is this a calculus question?
We using derivatives and such to figure this out?

Answer 4

yes for AP and i believe so

Answer 5

So we're given 2 pieces of information:
We have a constraint on our x and y,\[\Large\rm x+y=100,\qquad\qquad x,y\gt0\]And then we have a function of x and y,\[\Large\rm f(x,y)=x^2y\]That we want to maximize.

Answer 6

Before we can maximize the function,
We want to get it written in terms of only x, or only y,
so it'll be easier to work with.

Answer 7

So we'll use our constraint equation to get it in terms of only x, ok?

Answer 8

In our constraint equation we'll subtract x from each side,
\[\Large\rm \color{orangered}{y=100-x}\]We can use this to plug into our function,\[\Large\rm f(x,y)=x^2\color{orangered}{y}\]Do you understand how we'll plug it in? :o

Answer 9

okay

Answer 10

So plugging in the information:\[\Large\rm f(x)=x^2\color{orangered}{(100-x)}\]Gives us a function of only x, that we can differentiate with ease.
Before taking a derivative, expand out the brackets,\[\Large\rm f(x)=100x^2-x^3\]

Answer 11

To maximize this f function, you need to first find critical points.
Those points will correspond to max and min values of the function.

So to find critical points ( stationary points ), we set our first derivative equal to zero and solve for x.

\[\Large\rm f'(x)=?\]Do you understand how to find the derivative?

Answer 12

yes