let postive function $f(x)$ have two derivative on $(a,b),b>a$,and such $$f''(x)+f(x)\ge 0,x\in(a,b)$$ and $f(a)=f(b)=0$, show that

$$b-a\ge \pi$$

if $$f''(x)+f(x)=0$$ then $$f(x)=C_{1}\cos{x}+C_{2}\sin{x}$$ but this problem is inequality,so then I consider $$F(x)=f'^2(x)+f^2(x)\Longrightarrow F'(x)=2f'(x)f''(x)+2f(x)f'(x)=2f'(x)[f''(x)+f(x)]$$ But seem this also not solve this problem

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    $\begingroup$ Something is missing here. A constant function $f(x) = C > 0$ easily satisfies $f''(x) + f(x) \geq 0$; also, $f(x) = x^2 + 1$ on $(-\epsilon,\epsilon)$ is also a counterexample. $\endgroup$ – Pedro M. Jan 5 '15 at 13:29
  • $\begingroup$ sorry,I forgot $f(a)=f(b)=0$ $\endgroup$ – china math Jan 5 '15 at 13:30
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    $\begingroup$ @user201168: when the OP says that the function is defined on $(a,b)$, it is implicitly implied that $a < b$. $\endgroup$ – Pedro M. Jan 5 '15 at 13:40
  • $\begingroup$ this follows from sturm's comparison theorem. $\endgroup$ – abel Jan 5 '15 at 20:51

Assume that $b-a\lt\pi$ and consider $$c=\tfrac12(\pi-b-a),\qquad s=\sin(a+c).$$ Then $s=\sin(b+c)$ (always true), $s\gt0$ (because $0\lt b-a\lt\pi$), and $\sin(x+c)\gt0$ for every $x$ in $(a,b)$. Furthermore, $f(a)=f(b)=0$ and $f\gt0$ on $(a,b)$, hence $f'(a)\geqslant0$ and $f'(b)\leqslant0$, which implies that $f'(b)-f'(a)\leqslant0$.

All this being kept in store, let us consider the integral $$\int_a^b(f''(x)+f(x))\sin(x+c)\mathrm dx=\left.f'(x)\sin(x+c)-f(x)\cos(c+x)\right|_a^b=(f'(b)-f'(a))\,s. $$ The LHS is nonnegative as the integral of a nonnegative function and the RHS is nonpositive as the product of $f'(b)-f'(a)\leqslant0$ by $s\gt0$, hence the LHS is actually zero.

It follows that the function in the integral must be identically zero, that is, $f''+f=0$ on $(a,b)$. The rest is direct.


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