find local maximum/minima

how do i find the local maxima and minima for f(x,y)=110*x-4.5*x^2+115*y-2*y^2. this is my first time using matlab and i am not familiar with the function of the solver.

Answer 1

[Ameer Hamza]

Your function is Paraboloid, and it only has single minima or maxima.

If you have a symbolic toolbox, you can find the gradient and zero it to find the stationary point.

syms x y

f = 110*x-4.5*x^2+115*y-2*y^2;
df = gradient(f);
sol = solve(df)

Result

>> sol.x
ans =
110/9
>> sol.y
ans =
115/4

To check if it is minima or maxima, we can use hessian matrix

d2f = hessian(f)

Result

>> d2f = hessian(f)
d2f =
[-9,  0]
[ 0, -4]

The hessian is negative definite, therefore, the stationary point is maxima.

If you have optimization toolbox, then you can also try fmincon()

f = @(x,y) 110*x-4.5*x.^2+115*y-2*y.^2;
sol = fmincon(@(X) -f(X(1), X(2)), rand(2,1))