Can you help with question 1 here? al xm matrix. By the mean value theorem (Theorem 2.1.2), there exists a z in [x, y] such thatg(y) – 8 (x) = Vg(z) · (y – x) = b · (f'(z)(y – x)). The Cauchy-Schwarz inequality, (1.1.38), thenimplies |g(y) - 8 (x) < ||b| · |f'(z)V- )|| = ||f'(z)(y – x)||, and (3) follows.By applying the inequality (3) to the new function y f(y) -f'(w)y, with w a fixed vector, thefollowing inequality is obtained for all x, y, and w in R":|f(y) – f(x) – f'(w)(y – x)|| < max |(f'(z) – f'(w))(y - x)||(4)[A'x]ƏzPROBLEMS FOR SECTION 2.91. (a) Let f be defined for all (x, y) by f(x, y) =xy2and f (0, 0) = 0. Show that f{(x, y)and f;(x, y) exist for all (x, y).(b) Show thatf has a directional derivative in every direction at every point.(c) Show that f is not continuous at (0, 0). (Hint: Consider the behaviour of f along the curvex = y2.) Is f differentiable at (0, 0)?Existence and Uniqueness of Solutions ofSystems of Equations2.10This section is concerned with the svstem of n eauations in n unknowns, of the form10:18 P4/3/202BAdeletehomeOrpuaos ud->backspacewnulock8.6home

Name: Can you help with question 1 here? al xm matrix. By the mean value the
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Description: Can you help with question 1 here? al xm matrix. By the mean value theorem (Theorem 2.1.2), there exists a z in [x, y] such thatg(y) – 8 (x) = Vg(z) · (y – x) = b · (f'(z)(y – x)). The Cauchy-Schwarz inequality, (1.1.38), thenimplies |g(y) - 8 (x) <