A rectangle is inscribed in an isosceles triangle with base 1 unit and height h units. what should be the height of the triangle,\nso that it has maximum area?

There is some correction in question. The rectangle cant have any minimum area. We will find out the maximum area.
Let the triangle be ABC with BC on the x axis and point A on y axis.
A(0,h) B(-1/2,0) C(1/2,0)
Say the rectangle be EFGH with EF side on side BC.
E(-x,0) F(x,0) G(x,y) H(-x,y)
Now the points A G and C lie on same line. Hence, from slope
-2h = y/(x-1/2)
y = h - 2hx
Now the area of rectangle A = 2xy = 2xh(1-2x)
Differentiate it : 2h( 1 - 4x) = 0