# Integral of pointwise max function

Consider given some functions f1,f2,g∈C∞([a,b],R)f_1,f_2,g\in C^\infty([a,b],\mathbb{R}) which have both positive and negative values. Call F:[a,b]→RF:[a,b]\rightarrow \mathbb{R} the pointwise maximum of f1f_1 and f2f_2: F(x):=max{f1(x),f2(x)}F(x):=max\{f_1(x),f_2(x)\}. Suppose it is true ∫baf1gdx≥0\int_a^bf_1gdx\ge 0 and ∫baf2gdx≥0\int_a^bf_2gdx\ge 0. Then is it always true ∫baFgdx≥0\int_a^bFgdx\ge 0? I suspect the answer is yes, but I can’t work out a formal proof.

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Use the fact that max{f1,f2}=(f1+f2+|f1−f2|)/2\max\{ f_1, f_2\} = (f_1 + f_2 + |f_1 -f_2|)/2.
– ec92
2 days ago

thank you, but then what should I do with ∫ba|f1−f2|gdx\int_a^b|f_1-f_2|gdx?
– User29983
2 days ago

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