Question

1) Tim GTNN (x > 0)   B = \(5x+\frac{1}{x^3}\)

2) Rút gọn biểu thức  A = \(\sqrt{x+4\sqrt{x-4}}+\sqrt{x-4\sqrt{x-4}}\)

Answer 1

A = \(\sqrt{x-4+4\sqrt{x-4}+4}+\sqrt{x-4-4\sqrt{x-4}+4}\)

  = \(\sqrt{\left(\sqrt{x-4}+2\right)^2}+\sqrt{\left(\sqrt{x-4}-2\right)^2}\)

  = \(\sqrt{x-4}+2+l\sqrt{x-4}-2l\)

(+) với \(l\sqrt{x-4}-2l=\sqrt{x-4}-2\) khi \(x\ge8\)

=> A = \(\sqrt{x-4}+2+\sqrt{x-4}-2=2\sqrt{x-4}\)

(+) \(l\sqrt{x-4}-2l=2-\sqrt{x-4}\) khi \(4\le x\le8\)

=> A = \(\sqrt{x-4}+2+2-\sqrt{x-4}=4\)

 

Answer 2

1) Áp dụng bất đẳng thức Cô - si với 4 số \(\frac{5x}{3};\frac{5x}{3};\frac{5x}{3};\frac{1}{x^3}\) dương ta có:

 \(B=\frac{5x}{3}+\frac{5x}{3}+\frac{5x}{3}+\frac{1}{x^3}\ge4\sqrt[4]{\frac{5x}{3}.\frac{5x}{3}.\frac{5x}{3}.\frac{1}{x^3}}=4\sqrt[4]{\frac{125}{27}}\)

=> B nhỏ nhất bằng \(4\sqrt[4]{\frac{125}{27}}\) khi \(\frac{5x}{3}=\frac{1}{x^3}\) => x⁴ = 3/5 => x = \(\sqrt[4]{\frac{3}{5}}\)

2) ĐK : x > 4

\(A=\sqrt{\left(x-4\right)+2\sqrt{x-4}.2+4}+\sqrt{\left(x-4\right)-2\sqrt{x-4}.2+4}\)

\(A=\sqrt{\left(\sqrt{x-4}+2\right)^2}+\sqrt{\left(\sqrt{x-4}-2\right)^2}\)

\(A=\sqrt{x-4}+2+\left|\sqrt{x-4}-2\right|\)

+) Nếu \(\sqrt{x-4}\ge2\) => x - 4 > 4 => x > 8 thì \(A=\sqrt{x-4}+2+\sqrt{x-4}-2=2\sqrt{x-4}\)

+) Nếu \(\sqrt{x-4}