(a) Prove that for all x > 0, 1 x 2 * 1 1 x2 , ex2/2 < 1
Chapter , Problem 31(choose chapter or problem)
(a) Prove that for all x > 0, 1 x 2 * 1 1 x2 , ex2/2 < 1 -(x) < 1 x 2 ex2/2 . Hint: Integrate the following inequalities: (1 3y4 )ey2/2 < ey2/2 < (1 + y2 )ey2/2 . (b) Use part (a) to prove that 1 -(x) 1 x 2 ex2/2 . That is, as x , the ratio of the two sides approaches 1.
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