CMR (ax+by+cz)^2 $\leq$ (a^2+b^2+c^2)(x^2+y^2+z^2) 06/09/2021 Bởi Anna CMR (ax+by+cz)^2 $\leq$ (a^2+b^2+c^2)(x^2+y^2+z^2)
(ax+by+cz)² ≤ (a²+b²+c²)(x²+y²+z²) <=> (ax)²+ (by)²+ (cz)²+ 2axby+ 2bycz+ 2axcz ≤ (ax)²+(ay)²+(az)²+(bx)²+(by)²+(bz)²+(cx)²+(cy)²+(cz)² <=> 2axby+ 2bycz+ 2axcz ≤ (ay)²+(az)²+(bx)²+(bz)²+(cx)²+(cy)² <=> (ay)²+(az)²+(bx)²+(bz)²+(cx)²+(cy)²- 2axby- 2bycz – 2axcz ≥ 0 <=> (ay)² – 2axby+ (bx)²+ (az)²- 2axcz+ (cx)²+ (bz)²- 2bycz +(cy)² ≥0 <=> (ay- bx)²+ (az- cx)²+ (bz- cy)²≥0 (luôn đúng) =>đpcm P/s: Đây là BĐT bunhiacopxki Bình luận
Đây là BĐT Bunhiacopxki nha bạn Cm: Xét hiệu: $(a^2+b^2+c^2)(x^2+y^2+z^2)-(ax+by+cz)^2$ $=a^2x^2+a^2y^2+a^2z^2+b^2x^2+b^2y^2+b^2z^2+c^2x^2+x^2y^2+c^2z^2-a^2x^2-b^2y^2-c^2z^2-2axby-2axcz-2bycz$ $=a^2y^2+a^2z^2+b^2x^2+b^2z^2+c^2x^2+x^2y^2-2axby-2axcz-2bycz$ $=(a^2y^2-2axby+b^2x^2)+(a^2z^2-2axcz+c^2x^2)+(b^2z^2-2byxz+x^2y^2)$ $=(ay-bx)^2+(az-cx)^2+(bz-xy)^2≥0$ Bình luận
(ax+by+cz)² ≤ (a²+b²+c²)(x²+y²+z²)
<=> (ax)²+ (by)²+ (cz)²+ 2axby+ 2bycz+ 2axcz ≤ (ax)²+(ay)²+(az)²+(bx)²+(by)²+(bz)²+(cx)²+(cy)²+(cz)²
<=> 2axby+ 2bycz+ 2axcz ≤ (ay)²+(az)²+(bx)²+(bz)²+(cx)²+(cy)²
<=> (ay)²+(az)²+(bx)²+(bz)²+(cx)²+(cy)²- 2axby- 2bycz – 2axcz ≥ 0
<=> (ay)² – 2axby+ (bx)²+ (az)²- 2axcz+ (cx)²+ (bz)²- 2bycz +(cy)² ≥0
<=> (ay- bx)²+ (az- cx)²+ (bz- cy)²≥0 (luôn đúng)
=>đpcm
P/s: Đây là BĐT bunhiacopxki
Đây là BĐT Bunhiacopxki nha bạn
Cm: Xét hiệu: $(a^2+b^2+c^2)(x^2+y^2+z^2)-(ax+by+cz)^2$
$=a^2x^2+a^2y^2+a^2z^2+b^2x^2+b^2y^2+b^2z^2+c^2x^2+x^2y^2+c^2z^2-a^2x^2-b^2y^2-c^2z^2-2axby-2axcz-2bycz$
$=a^2y^2+a^2z^2+b^2x^2+b^2z^2+c^2x^2+x^2y^2-2axby-2axcz-2bycz$
$=(a^2y^2-2axby+b^2x^2)+(a^2z^2-2axcz+c^2x^2)+(b^2z^2-2byxz+x^2y^2)$
$=(ay-bx)^2+(az-cx)^2+(bz-xy)^2≥0$