Calculus problem

[Guest]

The following function is y = e to the minux x squared

The rectangle PQRS is found as above. P and Q are meant to be found at coordinates within this function.

My job is to find the value of x for which the rectangle has a maximum area and hence to compute the maximum areas of the rectangle...


My working out:

I am thinking that points P and Q are points of inflexion...hence by find the 2nd derivative and letting equal to 0 I can found the points R and S and
to find P and Q just substitute back into the y = function(x).

Am I on the right track?

TIA

 

[soroban]

Hello, americo74!

The following function is \(\displaystyle y\:=\:e^{-x^2}\)

                              |
                              *
                             *|*
                           *  |  *
                    P   *     |     *    Q
                    *---------+---------*
               *    |         |         |    *
         *          |         |         |y         *
  *                 |         |         |                   *
- - - - - - - - - - *---------+---------* - - - - - - - - - - -
                    S    x    |    x    R

Find the value of x for which the inscribed rectangle has maximum area
and find the maximum area of the rectangle.

The base of the rectangle is \(\displaystyle \,2x\).
The height of the rectangle is \(\displaystyle \,y\).

Hence, the area of the rectangle is: \(\displaystyle \,A\;=\;2xy\)

Since \(\displaystyle y\:=\:e^{-x^2}\), we have: \(\displaystyle \L\,A\;=\;2x\cdot e^{-x^2}\)

And that is the function you must maximize.

 

[Guest]

thank you Soroban you are a star!

a mathematical star!