Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 303 / 1 / a / i

\frac{\partial f}{\partial m} = 4 a_{4} m^{3} + 6 a_{6} m^{5} - B = 0,

and a local minimum must satisfy

\frac{\partial ^{2} f}{\partial m ^{2}} = 12 a_{4} m^{2} + 30 a_{6} m^{4} \geq 0.

One compares the value of

f

at every such local minimum and chooses the smallest. Since

a_{6} > 0

, the polynomial tends to positive infinity as

∣ m ∣

tends to infinity, so a global minimum exists.

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