ĐK: \(x\ge0\)

+) Với x = 0 => A = 0

+) Với x khác 0

Ta có: \(\frac{1}{A}=\frac{3}{4}\sqrt{x}-\frac{3}{4}+\frac{3}{4\sqrt{x}}=\frac{3}{4}\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)-\frac{3}{4}\ge\frac{3}{4}.2-\frac{3}{4}=\frac{3}{4}\)

=> \(A\le\frac{4}{3}\)

Dấu "=" xảy ra <=> \(\sqrt{x}=\frac{1}{\sqrt{x}}\)<=> x = 1

Vậy max A = 4/3 tại x = 1

Còn có 1 cách em quy đồng hai vế giải đenta theo A thì sẽ tìm đc cả GTNN và GTLN 

Bạn kiểm tra lại xem đã viết đúng đề chưa vậy?

\(\frac{3}{\sqrt{7}-1}+\frac{3}{\sqrt{7}+1}=\frac{3\left[\sqrt{7}+1+\sqrt{7}-1\right]}{\left(\sqrt{7}+1\right)\left(\sqrt{7}-1\right)}=\frac{6\sqrt{7}}{6}=\sqrt{7}\)

\(\frac{3}{\sqrt{X}-1}-\frac{2}{\sqrt{X}+1}+\frac{X-7}{X-1}=\frac{3\left(\sqrt{X}+1\right)-2\left(\sqrt{X}-1\right)+X-7}{\left(\sqrt{X}+1\right)\left(\sqrt{X}-1\right)}=\frac{X+\sqrt{X}-2}{\left(\sqrt{X}+1\right)\left(\sqrt{X}-1\right)}=\frac{\sqrt{X}+2}{\sqrt{X}+1}\)

TÍNH GIÁ TRỊ BIỂU THỨC:

\(\frac{3}{\sqrt{7}-1}\) + \(\frac{3}{\sqrt{7}+1}\)= \(\frac{3\left(\sqrt{7}+1\right)+3\left(\sqrt{7}-1\right)}{\left(\sqrt{7}-1\right)\left(\sqrt{7}+1\right)}\)= \(\frac{3\sqrt{7}+3+3\sqrt{7}-3}{6}\)=\(\frac{6\sqrt{7}}{6}\)=\(\sqrt{7}\)

RÚT GỌN BIỂU THỨC:

\(\frac{3}{\sqrt{X}-1}\)-\(\frac{2}{\sqrt{X}+1}\)+\(\frac{X-7}{X-1}\)

= \(\frac{3\left(\sqrt{X}+1\right)}{\left(\sqrt{X}-1\right)\left(\sqrt{X}+1\right)}\)-\(\frac{2\left(\sqrt{X}-1\right)}{\left(\sqrt{X}-1\right)\left(\sqrt{X}+1\right)}\)+\(\frac{X-7}{\left(\sqrt{X}-1\right)\left(\sqrt{X}+1\right)}\)

= \(\frac{3\sqrt{X}+3-2\sqrt{X}+2+X-7}{\left(\sqrt{X}-1\right)\left(\sqrt{X}+1\right)}\)

= \(\frac{X+\sqrt{X}-2}{\left(\sqrt{X}-1\right)\left(\sqrt{X}+1\right)}\)

= \(\frac{\left(\sqrt{X}+1\right)\left(\sqrt{X}-2\right)}{\left(\sqrt{X}-1\right)\left(\sqrt{X}+1\right)}\)

= \(\frac{\sqrt{X}-2}{\sqrt{X}-1}\)

CHÚC EM HỌC TỐT!

Đk: \(2\le x\le4\)

Áp dụng BĐT bunhiacopxki có:

\(P^2=\left(\sqrt{x-2}+3\sqrt{4-x}\right)^2\le\left(1+3^2\right)\left(x-2+4-x\right)\)

\(\Leftrightarrow P^2\le20\)\(\Leftrightarrow P\le2\sqrt{5}\)

Dấu "=" xảy ra khi \(\sqrt{x-2}=\dfrac{\sqrt{4-x}}{3}\) \(\Leftrightarrow x=\dfrac{11}{5}\) (tm đk)

Có \(P^2=8\left(4-x\right)+6\sqrt{\left(x-2\right)\left(4-x\right)}+2\ge2\)\(\Rightarrow P\ge\sqrt{2}\)

Dấu "=" xảy ra khi x=4 (tm)

\(a,P=\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)+1\)

\(=\left(x+1\right)\left(x+4\right)\left(x+2\right)\left(x+3\right)+1\)

\(=\left(x^2+5x+4\right)\left(x^2+5x+6\right)+1\)

\(=\left(x^2+5x+5-1\right)\left(x^2+5x+5+1\right)+1\)

\(=\left(x^2+5x+5\right)^2-1+1\)

\(=\left(x^2+5x+5\right)^2\ge0\forall x\)

Vậy \(P\ge0\forall x\)

\(b,P=\left(x^2+5x+5\right)^2\left(cmt\right)\)

Thay \(x=\frac{\sqrt{7}-5}{2}\)vào P ta được

\(P=\left(\left(\frac{\sqrt{7}-5}{2}\right)^2+5.\frac{\sqrt{7}-5}{2}+5\right)^2\)

\(=\left(\frac{7-10\sqrt{7}+25}{4}+\frac{10\sqrt{7}-50}{4}+\frac{20}{4}\right)^2\)

\(=\left(\frac{32-10\sqrt{7}+10\sqrt{7}-50+20}{4}\right)^2\)

\(=\left(\frac{2}{4}\right)^2\)

\(=\frac{1}{4}\)

a,

P=(x+1)(x+2)(x+3)(x+4)+1

P=[(x+1).(x+4)].[(x+2).(x+3)]+1

P=(x^2+5x+4)(x^2+5x+6)+1

P=[(x^2+5x+5)-1].[(x^2+5x+5)+1]+1

P=(x^2+5x+5)^2-1+1

P=\(\left(x^2+5x+5\right)^2\) \(\ge\)0 với mọi x

Câu b thì thay x vào rồi bấm máy ra ra kết quả

a, ĐK \(\hept{\begin{cases}x>0\\x\ne1\end{cases}}\)

\(P=\frac{x-1}{\sqrt{x}}:\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)+1-\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+1\right)}\)

\(=\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\sqrt{x}}.\frac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}=\frac{\left(\sqrt{x}+1\right)^2}{\sqrt{x}-1}\)

Ta thấy \(P=\frac{\left(\sqrt{x}+1\right)^2}{\sqrt{x}-1}>0\forall x>0,x\ne1\)

b, P=\(\frac{x+2\sqrt{x}+1}{\sqrt{x}-1}=\frac{\frac{2}{2+\sqrt{3}}+2\sqrt{\frac{2}{2+\sqrt{3}}}+1}{\sqrt{\frac{2}{2+\sqrt{3}}}-1}\)

=\(\frac{\frac{4}{\left(\sqrt{3}+1\right)^2}+2.\sqrt{\left(\frac{2}{\left(\sqrt{3}+1\right)^2}\right)}+1}{\sqrt{\left(\frac{2}{2+\sqrt{3}}\right)^2}-1}=\frac{\frac{4}{\left(\sqrt{3}+1\right)^2}+2.\frac{2}{\sqrt{3}+1}+1}{\frac{2}{\sqrt{3}+1}-1}\)

\(=\frac{12+6\sqrt{3}}{1-3}=-6-3\sqrt{3}\)

cậu ơi câu c đâu ạ??

Em không biết làm

a. x khac 1

b. 5-2√5 / 5

bạn làm chi tiết hơn đc không