- Bài này phải có điều kiện \(x>0\) thì mới làm được nhé bạn.

\(P=\dfrac{x^3+2021}{x}=x^2+\dfrac{2021}{x}=x^2+\dfrac{2021}{2x}+\dfrac{2021}{2x}\)

Áp dụng bất đẳng thức Cauchy cho 3 số dương ta có:

\(x^2+\dfrac{2021}{2x}+\dfrac{2021}{2x}\ge3\sqrt[3]{x^2.\dfrac{2021}{2x}.\dfrac{2021}{2x}}=3\sqrt[3]{\dfrac{2021^2}{4}}\)

Dấu "=" xảy ra khi \(x^2=\dfrac{2021}{2x}\Leftrightarrow x=\sqrt[3]{\dfrac{2021}{2}}\)

Vậy \(MinP=3\sqrt[3]{\dfrac{2021^2}{4}}\)

a: A=-2/15*25*x^2y*x^2y^4=-10/3*x^4y^5

B=2*(-5/6)*xy^3*x^3y=-5/3*x^4y^4

b: C=A+B*y

=-10/3*x^4y^5-5/3*x^4y^5

=-5x^4y^5

c: Khi x=2 và y=-1 thì C=-5*2^4*(-1)^5=5*16=80

1, \(\dfrac{a+b-c}{c}=\dfrac{a+c-b}{b}=\dfrac{b+c-a}{a}\)

 =>   \(\dfrac{a+b}{c}-1=\dfrac{a+c}{b}-1=\dfrac{b+c}{a}-1\)

 =>   \(\dfrac{a+b}{c}=\dfrac{a+c}{b}=\dfrac{b+c}{a}\)

=>    \(\dfrac{a+b}{c}=\dfrac{a+c}{b}=\dfrac{b+c}{a}=\dfrac{a+b+a+c+b+c}{a+b+c}=\dfrac{2\left(a+b+c\right)}{a+b+c}=2\)

=>  \(M=\dfrac{\left(a+b\right)\left(b+c\right)\left(c+a\right)}{abc}=\dfrac{a+b}{c}\times\dfrac{a+c}{b}\times\dfrac{b+c}{a}=2.2.2=8\)

=>   \(M=8\)

Bài 1:

Nếu $a+b+c=0$ thì đkđb thỏa mãn

$M=\frac{(-c)(-a)(-b)}{abc}=\frac{-(abc)}{abc}=-1$

Nếu $a+b+c\neq 0$. Áp dụng TCDTSBN:

$\frac{a+b-c}{c}=\frac{a+c-b}{b}=\frac{b+c-a}{a}=\frac{a+b-c+a+c-b+b+c-a}{c+b+a}=\frac{a+b+c}{a+b+c}=1$

$\Rightarrow a+b-c=c; a+c-b=b; b+c-a=a$

$\Leftrightarrow a+b=2c; a+c=2b; b+c=2a$

$\Rightarrow a=b=c$

$M=\frac{(a+a)(a+a)(a+a)}{aaa}=\frac{8a^3}{a^3}=8$

Bài 2a

Đặt $2x=3y=4z=t$

$\Rightarrow x=\frac{t}{2}; y=\frac{t}{3}; z=\frac{t}{4}$

Khi đó:

$|x+y+3z|=1$

$\Leftrightarrow |\frac{t}{2}+\frac{t}{3}+\frac{3t}{4}|=1$

$\Leftrightarrow |\frac{19}{12}t|=1$

$\Rightarrow t=\pm \frac{12}{19}$

Nếu $t=\frac{12}{19}$ thì:

$x=\frac{t}{2}=\frac{6}{19}; y=\frac{4}{19}; z=\frac{3}{19}$

Nếu $t=-\frac{12}{19}$ thì:

$x=\frac{t}{2}=\frac{-6}{19}; y=\frac{-4}{19}; z=\frac{-3}{19}$

Bài 2b:

$\frac{x^3}{8}=\frac{y^3}{27}=\frac{z^3}{64}$

$\Leftrightarrow (\frac{x}{2})^3=(\frac{y}{3})^3=(\frac{z}{4})^3$

$\Leftrightarrow \frac{x}{2}=\frac{y}{3}=\frac{z}{4}$

Đặt $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}=t$

$\Rightarrow x=2t; y=3t; z=4t$

Khi đó:
$x^2+2y^2-3z^2=-650$

$\Leftrightarrow (2t)^2+2.(3t)^2-3(4t)^2=-650$

$\Leftrightarrow -26t^2=-650$

$\Leftrightarrow t=\pm 5$

Nếu $t=5$ thì:

$x=2t=10; y=3t=15; z=4t=20$

Nếu $t=-5$ thì:

$x=2t=-10; y=3t=-15; z=4t=-20$