2. Let f(r) = In(1 + z). Which of the following statements are true? (A) The Taylor series about o = O js x-x2 + x3-x4+ .... for |x |＜1. (B) The Taylor series about t = O is x -x2/2 + x3/3-x4/4+..., for |x |＜1. (C) f'(x) is increasing in z for r ＞ -1. (D) f"(x) is increasing in a for x > -1. (E) None of the above.

複選題10. Let A be an n x n matrix whose (i, j) component is aij. Let f be a real-valued function defined on A. Let ∇Af( A ) be the gradient of f( A ) with respect to A； ∇Af( A )is delined as an n x n matrix whose (i, j) entry is . Let B and C be n x n matrices. Which of the following statements are true? (A) If f( A ) = tr(AB), then ∇Af( A )= B. (B) If f( A ) = tr(AB), then ∇Af( A ) = BT. (C) If f( A ) = tr(AATC), then ∇Af( A ) = CA + CTAT. (D) If f( A ) == tr(AATC), then ∇Af( A ) = CA + CAT. (E) None of the above.

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