Question

If x > 0, then the minimum value of $x^{1000}+x^{900}+x^{90}+x^6+\frac{1996}{x}$ is

• A. 1000
• B. 2000
• C. 1996
• D. 3000

Answer 1

Answer: (B)

2000

$x^{1000}+x^{900}+x^{90}+x^6+\frac{1996}{x}$

splitting $\frac{1996}{x}$ into $\frac{1}{x}$ 1996 times and adding we get

as AM$\ge$GM

$\frac{x^{1000}+x^{900}+x^{90}+x^6+\frac{1}{x}+....+\frac{1}{x}}{2000}\ge 1$

Hence $x^{1000}+x^{900}+x^{900}+x^6+\frac{1996}{x}\ge 2000$