GRE Exam Numerical Ability Geometry

Let f(x, y) = x^2 – 2xy + y^3 for all real x and y. Which of the following is true.
(A) f has all of its relative extrema on the line x y.
(B) f has all of its relative extrema on the parabola x = y^2
(C) f has a relative minimum at (0, 0)
(D) f has an absolute minimum at (2/3, 2/3)
(E) f has an absolute minimum at (1, 1)

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GRE Other Question

For what value of b is the line y = 10 x tangent to the curve y = e^(bx) at some point in the xy-plance?
(A) 10/e
(B) 10
(C) 10 e
(D) e^10
(E) e Suppose that two binary operations, denoted by + and * are defined on a nonempty set S, and that the
following conditions are satisfied for all x, y, and z in S:
(1) x+y and x * y are in S
(2) x+(y+z) = (x+y)+z and x*(y*z) = (x*y)*z
(3) x+y = y+x
Also, for each x in S and for each positive integer n, the elements nx and n x are defined recursively as x^n are defined recursively as follos:
1x = x^1 = x and
If kx = x^k have been defined, then (k+1)x = kx + x and x^(k+1) = x^k * x.
Which of the following must be true?
I. (x*y)^n = x^n * y^n for all x and y in S and for each positive integers n.
II. n(x+y) = nx ny for all x and y in S and for each positive integer n.
III. x^m * x^n = x^(m+n) for each x in S and for all positive integers m and n.
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III