For x n 1 f x n f 00 x n 0 thus x n x n 1 x n moves

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For x n < 1, f 0 ( x n ) f 00 ( x n ) < 0 thus, x n < x n +1 : x n moves to the right for 1 < x n < 15 / 6, f 0 ( x n ) f 00 ( x n ) > 0 thus, x n > x n +1 : x n moves to the left for 15 / 6 < x n < 4, f 0 ( x n ) f 00 ( x n ) < 0 thus, x n < x n +1 : x n moves to the right for 4 < x n , f 0 ( x n ) f 00 ( x n ) > 0 thus, x n > x n +1 : x n moves to the left When x n = 1 or x n = 4, f 0 ( x n ) f 00 ( x n ) = 0 and x n = x n +1 , and finally the process ends. (d) (Ans) When f ( x ) is continuous and differentiable functions, it has (a) local minimum(s) x min when f 0 ( x min ) = 0 and f 00 ( x min ) > 0, and a local maximum(s) x max when f 0 ( x max ) = 0 and f 00 ( x max ) < 0. Note that in this example we have f 0 (1) = f 0 (4) = 0, and f 00 (1) < 0 and f 00 (4) > 0. We can say, f ( x ) has a local minimum at x = 4 and a local maxmum at x = 1. From the graph, we can see that there is no global minimum of global maximum. 3. (a) (Ans) Since we are interested in the expected change in man’s income, we start by setting F i = 0 . The regression equation becomes Y i = 21 . 3 + 2 . 5 X i + ˆ u i . 3
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Therefore the man’s income is expected to change by 2 . X i when there is a change in the variable X i . If he acquires an additional year of education (Δ X i = 1), we may expect that his income will rise by $2,500. (b) (Ans) This time we are interested in the expected change in woman’s income, the case when F i = 1. The regression equation becomes Y i = 21 . 3 + 2 . 7 X i - 5 . 2 F i + ˆ u i . If she acquires an additional uear of education (Δ X i = 1), her expected income will rise by $2,700. (c) (Ans) First, we notice that X i F i is the relevant variable to the difference in returns to education between gender. Therefore we perform the T-test to see if the coefficient of X i F i is zero or not. The t-statistic is t = 0 . 2 0 . 05 = 4 and we find that | t | > 1 . 96 = z 0 . 975 . Therefore we reject the null hypothesis at 95% significance level: there is a statistical evidence that the returnes to education differ between men and women.
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