The rectangle will have width 2x and height `4e^(-2x^2+3)` . (Note that the given function is symmetric about the y-axis.)
The area will be `A=(2x)(4e^(-2x^2+3))=8xe^(-2x^2+3)`
To maximize this function we take the first derivative and set it equal to zero in order to find the critical points:
`(dA)/(dx)=8e^(-2x^2+3)+(8x)(-4x)e^(-2x^2+3)` using the product rule
`=e^(-2x^2+3)[8-32x^2]` factoring out the common factor.
Setting this equal to zero we get:
`e^(-2x^2+3)[8-32x^2]=0 ==>8-32x^2=0` since e to a power is positive.
Thus the width of the rectangle is 1 (2*1/2) and the height is `f(1/2)=4e^(-2(1/2)^2+3)~~48.73`
The maximal area is approximately 48.73 square units.