 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2013/2/b/solution

Continuity at

x = 2

requires both one-sided limits to equal the specified value

3

. The right branch gives

a + 2 b = 3

and the left branch gives

b - 4 a = 3

. Solving these two linear equations gives

a = - \frac{1}{3}, b = \frac{5}{3} .

Substitution gives

- 1/3 + 10/3 = 3

on the right and

5/3 + 4/3 = 3

on the left. These conditions enforce continuity, without requiring the two one-sided derivatives to agree.

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