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Let $f: \Bbb R \to \Bbb R$ be a function such that $f'(x)$ exists and is continuous over $\Bbb R$. Moreover, let there be a $T > 0$ such that $f(x + T) = f(x)$ for all $x \in \Bbb R$ and let $f(x) + f'(x)\ge 0$ for all $x \in \Bbb R$.

Show that $f(x) \ge 0$ for all $x \in \Bbb R$.

My attempt: $f(x) \ge 0 \iff f(x) \ge f'(x) - f'(x) \iff f(x) + f'(x) \ge f'(x)$.

Thus, it is enoguh to show that $0 \ge f'(x)$.

$\iff 0 \ge \lim_{h\to0}\frac{f(x + h) - f(x)}{h}$

I do not know how to proceed from here. I know that $f'$ also has a periodicity of T but I do not know how to use that here.

Am I on the right track? How can I use the periodicity of $f$ to solve the problem?

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7 Answers 7

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Let $g(x)=\mathrm e^xf(x)$, then $g'(x)=\mathrm e^x(f'(x)+f(x))\geqslant0$ hence: $$(1)\ \textit{The function $g$ is nondecreasing.}$$ Since $f$ is continuous and periodic, $f$ is bounded, say $|f(x)|\leqslant C$ for every $x$, hence $|g(x)|\leqslant C\mathrm e^x\to0$ when $x\to-\infty$, that is: $$(2)\ \textit{The function $g$ has limit $0$ at $-\infty$.}$$ Properties (1) and (2) of $g$ imply together that $g\geqslant0$ everywhere, hence $f\geqslant0$ everywhere.


Examples: Consider $$f(x)=c\,\mathrm e^{w\cos(ux+v)},$$ for every $c\geqslant0$, $u\ne0$, $|uw|\leqslant1$ and $v$, then $f$ has period $2\pi/|u|$ and, for every $x$, $$f'(x)+f(x)=(1-wu\sin(ux+v))\,f(x)\geqslant0.$$ For example, if $c=w=u=1$, $v=0$, one gets the function $f$:

$\qquad\qquad\qquad$enter image description here

...And the function $f'+f$:

$\qquad\qquad\qquad$enter image description here

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We don't even need that $f'(x)$ is continuous! It is enough to show that $f(x) \ge 0$ on $[0,T]$, because of periodicity. Since $[0,T]$ is compact, $f$ has a minimum on $[0,T]$. Let $x_0 \in [0,T]$ be such that $f(x_0)$ is minimal. If $x_0 =0$ or $x_0 =T$, then $f(x_0)$ is also minimal in $[-T,2T]$, because of periodicity.

A necessary condition for a minimum is that $f'(x_0) =0$. (And it is also valid in the case $x_0 =0$ or $x_0 =T$, because of the above remark.) Thus we get $f(x_0) = f(x_0)+f'(x_0) \ge 0$. This shows that $f(x) \ge 0$ everywhere.

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  • $\begingroup$ You do need $f'(x)$ continuous when you say "a necessary condition for a minimum is that $f'(x_0)=0$". $\endgroup$ Nov 30, 2018 at 13:41
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    $\begingroup$ That is not true! For example, if $x_0$ is a minimum, then we have $f(x_0) \le f(x)$ for $|x-x_0| \le \delta$. Thus $f'(x_0) =\lim_{h \downarrow 0} (f(x_0+h)-f(x_0))/h \le 0$. On the other hand $f'(x_0) =\lim_{h \downarrow 0} (f(x_0-h)-f(x_0))(-h) \ge 0$. Thus $f'(x_0)=0$. In this (almost trivial) proof I haven't used coninuity of $f'$ and we don't need it! $\endgroup$
    – p4sch
    Nov 30, 2018 at 13:44
  • $\begingroup$ Yeah, I was thinking about $f(x)=|x|$, but in that case $f'(0)$ doesn't even exist. $\endgroup$ Dec 1, 2018 at 10:21
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Since $f$ is periodic, it has maximum and minimum. Choose the period $[a,b]$, such that $f(a)=f(b)=M$, where $M$ is the maximum.

It can be seen that, there is a point $f(c)=m$, where $m$ is the minimum. c is golbal minimum, and thus local minimum, so:

$$f'(c)=0$$

from your condition.

$$f+f'\ge 0$$

we see

$$f(c)\ge 0$$. Since $f(c)$ is the minimum, we are done!

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Consider such $f(x)$ to be non-constant. There must be a global minimum within each period, say at $x=x_0$. Since $f(x)$ is differentiable and continuous, $f'(x_0) = 0$. Hence $$f(x) \ge f(x_0) = f(x_0)+f'(x_0) \ge 0$$

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suppose $f(x)$ is negative for all values of $x$. Periodicity tells us that there must be solutions to $f'(x)=0$ and for such $x$ we would then have $f(x)+f'(x)=f(x)<0$.

Suppose that $f(x)$ is negative for some values of $x$ but not all. Let $a,b$ be zeroes of $f(x)$ such that $a<x<b\implies f(x)<0$. Specifically, let $x_0$ be some value for which $f(x_0)<0$ and let $a$ be the greatest upper bound of $\{x\,:\,x<x_0\,\;\&\,\;f(x)≥0\}$ and $b$ defined similarly on the other side. Since $f(a)=0=f(b)$ there is some $c$ with $a<c<b$ and $f'(c)=0$. But for that value we must have $$f(c)+f'(c)=f(c)<0$$ contrary to assumption.

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Say a period is $[a,a+T]$. Since $f$ is continuous, it attains a minimum on that interval, say at $c$. (If you have $c = a$, then change the period to $[a - T/2,a + T/2]$ so that $c$ becomes an interior point.)

We must have $f'(c) = 0$ there, so $f(c) \geq 0$. So the minimum value of $f$ is nonnegative.

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$ f \ne 0$

$f(x)=0 \Rightarrow f'(x) \ge 0$, hence $f$ can cross the X-axis at most once. periodicity means that it cannot cross at all.

if $f$ is non-positive then $f' \ge 0$ so $f$ is monotone increasing. again this contradicts periodicity

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  • $\begingroup$ Why is this downvoted? This answer is essentially saying if $f(x)<0$ somewhere, then $f(x)$ is strictly increasing there, contradicting the fact that $f(x)$ is periodic. $\endgroup$
    – peterwhy
    Nov 23, 2014 at 16:48

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