Find the elasticity of demand at the point on the demand fun

Find the elasticity of demand at the point on the demand function where total revenue is maximum.Please use first order and 2nd order maximum and minimum condition.Please take the demand function as p=f (q) and total revenue =pq

Solution

Total Revenue=f(q)*q

Function Revenue should be differentiated

R\'=f(q)+qf\'(q)...........Equation 1

R\"=f\'(q)+f\'(q)+qf\"(q)........Eqn 2

Equalting Equation 1 with 0 we get

f(q)=-qf\"(q)

q=-f(q)/f\"(q)

For Minimum R\">0 and for Maximum R\"<0

R\"=f\'(q)+f\'(q)+qf\"(q)=2f\'(q)+qf\"(q)

2f\'(q)-f(q)

dP/dQ-P<0 Because for Normal good dP/dQ symbol is negative always

Elasticity of Demand e=dq/q*f(q)/f\'(q)=dq*(f(q)/f\'(q))/q=-dq