Utility labor-leisure sum plz help

[man123]

Utility function between leisure L (hours/week) and earned income Y (Rs/week) of a person is given by
 U= 15LY – 100 L – 140 Y – 25L^2 – 2Y^2. Find the optimum values of L and Y and hence determine the implied wage rate

[Akshay Jain]

here we have to find the values of L and Y at which the function has a max value
find the following derivatives and then solve hessian matrix
dU/dL=0, d^2U/dL^2,
dU/dY=0, d^2U/dY^2,
d^2U/dYdL
hessian matrix = (d^2U/dL^2)*(d^2U/dY^2)-(d^2U/dYdL)^2
conditions for maxima
(d^2U/dL^2)<0
hessian matrix >0
optimum values L=100/17 Y=1340/51