Q220 · Practice questionTechnology-freeComplex familiar5 marks
QUESTION 220 (5 marks)
A positive solution satisfies and y(0)=1.
a)[3 marks]
Determine y explicitly.
b)[2 marks]
Show that this solution has a unique global minimum, and state it.
WORKED SOLUTION
5 marksPractice marking scheme
ANSWER
(a) . (b) Minimum at x=0.
Worked solution
(a) Separate as . Integration gives . The initial value gives C=2, so and the stated solution. The square-root expression stays positive for all real x.
(b) , so its square is at least 1. Equality occurs only at x=0, giving the unique global minimum.
Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Separate correctly.
Part a: Integrate and apply the initial value.
Part a: Give the positive explicit solution.
Part b: Establish the global lower bound.
Part b: Identify the sole equality point.
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