Find the real polynomial p(x) of degree 4 with largest possible coefficient of x4 such that p( [-1, 1] ) ∈ [0, 1].
Solution
Let p(x) = a4x4 + a3x3 + a2x2 + a1x + a0. Then a4x4 - a3x3 + a2x2 - a1x + a0 also satisfies the conditions and hence also a4x4 + a2x2 + a0. So we may take a3 = a1 = 0.
Now p(0), p(1/2), p(1) belong to [0, 1], so 0 ≤ a0 ≤ 1, 0 ≤ a4/4 + a2/2 + a0 ≤ 1, 0 ≤ a4 + a2 + a0 ≤ 1. Hence: -2 ≤ -2a0 ≤ 0, 0 ≤ a4 + 2a2 + 4a0 ≤ 4, -2 ≤ -2a4 - 2a2 - 2a0 ≤ 0. Adding: -4 ≤ -a4 < 4. So the maximum value of a4 is at most 4.
This can be achieved by 4(x2 - 1/2)2 = 4x4 - 4x2 + 1.
© John Scholes
jscholes@kalva.demon.co.uk
30 Nov 1999