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How to apply matrices, Jacobian, quasi-concavity, Hessian, Kuhn-Tucker conditions?

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The expert prove that quasi-concavity does not imply concavity.

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please refer to the attachment.

1. (a) Prove that quasi-concavity does not imply concavity.

The function f(x,y) defined by:

is quasi-concave, but not concave.

Here's the graph in two dimensions:

(b) Let ƒ be a C2 function on a convex subset D of R2. Assume that ƒ is strictly monotonic increasing. Prove that

(i) if the bordered Hessian determinant of ƒ is positive (negative) for all x Є D, then ƒ is quasi-concave (quasi-convex) on D; and

The bordered Hessian matrix for the function is
| f_11 f_12 ... f_1n f_1 |
| f_21 f_22 ... f_2n f_2 |
H = | ...

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