Conditions for function being concave, convex, max, min

[am99]

Hi Guys,

I am just a bit confused on what these conditions mean.

for the questions, such as for example find the curvature a function such as (x1+x2)^2.

I know that you take the first derivative and then the second-order derivative to build the hessian matrix ( I have attached the answer)
However, after this, I don't understand the conditions as to what makes it concave or convex, so if someone could explain the conditions it would really help me because I don't understand what H1 is? I know that H2 is the determinant of the hessian matrix.

 

[am99]

Hi,

I have worked out for when it is convex as its pretty straightforward because it is positive definite when both of its leading principal minors so H1 and H2 are positive ( H1 = F1), also it is positive semidefinite and convex when one of its leading principal minors are positive and the other is zero.

Does that sound right?

I'm still a bit confused on the concave ones as they don't seem to follow that same rule e.. H1 =negative, H2=negative, or semidefinite , H1 =negative H2=zero