A symmetric 2×2 matrix (i. e. T= ) is negative semi- definite, i. e. T≤0 for all ∈ℝ2 , if and only if both of the following is true: tr()≤0 det()≥0 (This fact can be explained in terms the eigenvalues of . Let 1 and 2 be the eigenvalues of , then tr()=1+2 while det()=12 . The two conditions above ensure that 1,2≤0 . ) Use the fact given above to determine whether the following functions concave, convex, or neither. fs(theta 1, theta 2) = - theta1^4 - theta 2^4 - (theta 2 - theta 1)^3
a. concave
b. convex
c. not concave and not convex