MathDB

8

Part of 2005 Junior Balkan Team Selection Tests - Moldova

Problems(1)

h (x) = \lambda f_m (x) + (1- \lambda) g_m (x) if g_m (x) <=h (x)<= f_m (x)

Source: 2005 Moldova JBMO TST p8

2/20/2021
The families of second degree functions fm,gm:R→R,f_m, g_m: R\to R, are considered , fm(x)=(m2+1)x2+3mx+m2−1f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1, gm(x)=m2x2+mx−1g_m (x) = m^2x^2 + mx - 1, where mm is a real nonzero parameter. Show that, for any function hh of the second degree with the property that gm(x)≤h(x)≤fm(x)g_m (x) \le h (x) \le f_m (x) for any real xx, there exists λ∈[0,1]\lambda \in [0, 1] which verifies the condition h(x)=λfm(x)+(1−λ)gm(x)h (x) = \lambda f_m (x) + (1- \lambda) g_m (x), whatever real xx is.
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