Question Number 55643 by gunawan last updated on 01/Mar/19
![known function f diferensiable continues at [a, b] If f(a)=f(b)=0 and ∫_a ^b [f(x)]^2 dx=1 Prove that ∫_a ^b x^2 [f′(x)]^2 dx ≥(1/4)](Q55643.png)
$$\mathrm{known}\:\mathrm{function}\:{f} \\ $$$$\mathrm{diferensiable}\:\mathrm{continues}\:\mathrm{at}\:\left[{a},\:{b}\right] \\ $$$$\mathrm{If}\:{f}\left({a}\right)={f}\left({b}\right)=\mathrm{0} \\ $$$$\mathrm{and}\: \\ $$$$\int_{{a}} ^{{b}} \left[{f}\left({x}\right)\right]^{\mathrm{2}} {dx}=\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\int_{{a}} ^{{b}} {x}^{\mathrm{2}} \left[{f}'\left({x}\right)\right]^{\mathrm{2}} \:{dx}\:\geqslant\frac{\mathrm{1}}{\mathrm{4}} \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 01/Mar/19
![[f(x)]←is [.] greatest integer function or simply bracket...](Q55647.png)
$$\left[{f}\left({x}\right)\right]\leftarrow{is}\:\left[.\right]\:{greatest}\:{integer}\:{function}\:{or}\: \\ $$$${simply}\:{bracket}... \\ $$