Đặt \(\left\{{}\begin{matrix}x-1=a>0\\y-1=b>0\end{matrix}\right.\)
\(P=\frac{\left(a+1\right)^2}{b}+\frac{\left(b+1\right)^2}{a}\ge\frac{\left(a+b+2\right)^2}{a+b}=\frac{\left(a+b\right)^2+4\left(a+b\right)+4}{a+b}\)
\(P\ge a+b+\frac{4}{a+b}+4\ge2\sqrt{\frac{4\left(a+b\right)}{a+b}}+4=8\)
\(P_{min}=8\) khi \(a=b=1\) hay \(x=y=2\)
1) Cho bieu thuc: \(B=\left(\frac{\sqrt{x}}{\sqrt{x}+4}+\frac{4}{\sqrt{x}-4}\right):\frac{x+16}{\sqrt{x}+2}\left(x\ge0,x\ne16\right)\)
a) Cho bieu thuc A= \(\frac{\sqrt{x}+4}{\sqrt{x}+2}\) ; voi cac cua bieu thuc A va B da cho, hay tim cac gia tri cua x nguyen de gia tri cua bieu thuc B(A;-1) la so nguyen
tim gia tri nho nhat cua bieu thuc P=\(\left(1+x\right)\left(1+\dfrac{1}{y}\right)+\left(1+y\right)\left(1+\dfrac{1}{x}\right)\) trong do x,y la cac so duong thoa man \(x^2+y^2=1\)
Cho cac so thuc duong x,y,z thoa man :\(\sqrt{x^2+y^2}+\sqrt{y^2+z^2}+\sqrt{z^2+x^2=2015}\)
Tim ja tri nho nhat cua bieu thuc :\(T=\dfrac{x^2}{y+x}+\dfrac{y^2}{z+x}+\dfrac{z^2}{x+y}\)
1) Cho bieu thuc \(A=\frac{x}{x-4}+\frac{1}{\sqrt{x}-2}+\frac{1}{\sqrt{x}+2}\)
a) Tim tat ca cac gia tri cua x de A>1
Cho a,b,c la cac so duong thoa man a+b+c=9.Tim gia tri nho nhat cua bieu thuc:
\(P=a^2+\frac{1}{a^2}+b^2+\frac{1}{b^2}+c^2+\frac{1}{c^2}\)
tim GTNN va GTLN cua bieu thuc D=\(\frac{4x+3}{x^3+1}\)
cho bieu thuc A = \(\left(\frac{x-\sqrt{x}}{\sqrt{x}-1}+1\right):\left(\frac{x+\sqrt{x}}{\sqrt{x}+1}\right)\)
a. tim x de bieu thuc A co nghia ?rut gon A ?
b. tinh gia tri cua bieu thuc A tai x=7+4√3
Cho 3 so x, y, z thoa man cac he thuc: \(\left(z-1\right)x-y=1\) va \(x+zy=2\)
Chmr: \(\left(2x-y\right)\left(z^2-z+1\right)=7\) va tim tat ca cac so nguyen x, y, z thoa man cac he thuc tren.
cho bieu thuc A =\(\left(\frac{x-\sqrt{x}}{\sqrt{x}-1}+1\right):\left(\frac{x+\sqrt{x}}{\sqrt{x}+1}\right)\)
(x≥0;x≠1)
a. tim x de bieu thuc A co nghia ?rut gon A ?
b. tinh gia tri bieu thuc A tai x=7+4√3