Question Number 215020 by mnjuly1970 last updated on 26/Dec/24
![f: [0 , 1] →R is given. f ′′ is continuous . by the way f(0)=f(1). prove that : determinant (((∫_0 ^( 1) ( f ′′ (x))^( 2) dx ≥ 3(f ′(1))^2 )))](https://www.tinkutara.com/question/Q215020.png)
$$ \\ $$$$\:\:\:\:{f}:\:\:\left[\mathrm{0}\:,\:\mathrm{1}\right]\:\rightarrow\mathbb{R}\:{is}\:{given}. \\ $$$$\:\:\:\:{f}\:''\:\:\:\:{is}\:{continuous}\:. \\ $$$$\:\:\:\:{by}\:{the}\:{way}\:\:{f}\left(\mathrm{0}\right)={f}\left(\mathrm{1}\right). \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\:\: \\ $$$$\begin{array}{|c|}{\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\:{f}\:''\:\left({x}\right)\right)^{\:\mathrm{2}} {dx}\:\geqslant\:\mathrm{3}\left({f}\:'\left(\mathrm{1}\right)\right)^{\mathrm{2}} }\\\hline\end{array} \\ $$$$ \\ $$
Commented by MathematicalUser2357 last updated on 26/Dec/24
minjuly1970❤it's been 2~3 months
Commented by mnjuly1970 last updated on 26/Dec/24
$$\:\underbrace{\lesseqgtr} \\ $$
Answered by MrGaster last updated on 26/Dec/24
$$\left.\int_{\mathrm{0}} ^{\mathrm{1}} \left(\int^{''} \left({x}\right)\right)\right)^{\mathrm{2}} {dx}\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}} {dx}\geq\left(\int_{\mathrm{0}} ^{\mathrm{1}} {xf}^{''} \left({x}\right){dx}\right)^{\mathrm{2}} \\ $$$$\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{1}} \left({f}''\left({x}\right)\right)^{\mathrm{2}} {dx}\geq\left({f}'\left(\mathrm{1}\right)\right)^{\mathrm{2}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left({f}''\left({x}\right)\right)^{\mathrm{2}} {dx}\geq\mathrm{3}\left({f}'\left(\mathrm{1}\right)\right)^{\mathrm{2}} \\ $$
Commented by mnjuly1970 last updated on 27/Dec/24
$$\:\:\:\:\underbrace{\lesseqgtr} \\ $$