The region R is a rectangle with vertices P alna Q a0 S 3 0

The region R is a rectangle with vertices P (a,lna), Q (a,0), S (3, 0), and T (3, lna), where 1 < a< 3. The area of rectangle is maximized for some c between 1 and 3.

Write the expression you would need to solve in order to find c. You don\'t have to find c.

Solution

we have to maximise the area

so first write the expression for area

\\length of side PQ = lna

length of side QS = a-3

hence area=(a-3)lna

for some c area will be

A =(c-3)lnc

To maximise area find derivative of A and equate it to 0

dA/dc = (c-3)/c + lnc = 0

solve this for c and make sure that c lies between 1 and 3