Question

find the dimensions of the largest rectangle that can be inscribe in the region bounded by f(x)=-x^2+4 and g(x)=x^2-4

Answer 1

\[
Area(x)= 2 x ( 2(-x^2 +4)=16 x-4 x^3\\
Area'(x)=16 -12 x^2\\
Area'(x)= 0 \text{ for } x= \frac{2}{\sqrt{3}} \text { and } x=-\frac{2}{\sqrt{3}}\\
Area''( \frac{2}{\sqrt{3}})= -16 \sqrt{3} <0\\

\]
The dimensions are;;
\[ 2 \frac{2}{\sqrt{3}} = \frac{4}{\sqrt{3}}\\
2 ( -\left(\frac{2}{\sqrt{3}})\right)^2 +4=\frac {16} 3

\]