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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1

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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1 Answer

1

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The claim is false. As a counterexample, let [a,b]=[−1,1][a,b] = [-1,1], and

g(x)=−1f1(x)={−2x<01x≥0f2(x)={1x<0−2x≥0.\begin{align*}
g(x) &= -1 \\
f_1(x) &=
\begin{cases}
-2 & x \lt 0 \\
1 & x \ge 0
\end{cases} \\
f_2(x) &=
\begin{cases}
1 & x \lt 0 \\
-2 & x \ge 0
\end{cases}
\text{.}
\end{align*}
Observe that ∫[−1,1]f1g=∫[−1,1]f2g=1\int_{[-1,1]} f_1 g = \int_{[-1,1]} f_2 g = 1, but ∫[−1,1]Fg=∫[−1,1]−1=−2\int_{[-1,1]} F g = \int_{[-1,1]} -1 = -2.
Now, I know you asked for a smooth functions, and that f1f_1 and f2f_2 are not smooth. However, the fundamental problem is present with or without the smoothness requirement (for any ϵ\epsilon, you can convert these into smooth functions without changing the relevant integrals by more than ϵ\epsilon), and it's easiest to see for piecewise defined functions.
thank you, but what if both f1f_1 and f2f_2 are positive functions?
– User29983
2 days ago
@User29983 Thank you for your interest in my answer. In the future, please refrain from editing questions to significantly change their meaning, especially after they've received answers. This policy is out of consideration for the people who take the time to answer questions, to ensure that their efforts aren't spent chasing moving goalposts. If you'd like to ask a new question, you should do so with an actual new question on the site. You can even put a link to this question in your new question and say that you needed an additional assumption to get what you were looking for.
– Mike Haskel
2 days ago
@User29983 See, for example this post on the meta-site.
– Mike Haskel
2 days ago
1
Take for example [a,b]=[−π,2π][a,b] = [-\pi, 2\pi] with g(x)=sinxg(x) = \sin x, let f1f_1 be 11 on [−π,π][-\pi,\pi] and 00 on (π,2π](\pi,2\pi] and let f2f_2 be 00 on [−π,0)[-\pi,0) and 11 on [0,2π][0,2\pi]. Smooth it. (add a constant if you want strictly positive f1,f2f_1,f_2)
– Daniel Fischer♦
2 days ago