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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 5A / iii / Solution

Take

q_{1} (x) = 1, φ_{1} (x) = sin (x + 1) .

Its consecutive zeros are

- 1

and

π - 1

. For the stated equation,

q_{2} (x) = 2 + cos 3 x \geq 1 = q_{1} (x),

and

q_{2}

is not identically equal to

1

on the interval. The Sturm comparison theorem therefore shows that every nontrivial solution has at least one zero in

[- 1, π - 1] .

Solved by gpt-5.6-sol high.

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