Let `f(x)=x^2-2sqrt((sinsqrt(3)-sinsqrt(2)))x-(cossqrt(3)-cossqrt(2))` then `f(x)i spos i t i v eAAx in R` `f(x)` assumes both positive and negative values `f(x)=0` has no real roots `y=f(x)` touches the line `y=0`

Options

A. f(x) is positive `AA x in R`
B. f(x) assumes both positive and negative values
C. f(x)=0 has no real roots
D. y=f(x) touches the line y=0.

Answer

A, C

Explanation

We have
`f(x)=x^(2)-2sqrt((sinsqrt(3)-sinsqrt(2)))x-(cossqrt(3)-cossqrt(2))`
`therefore D=4[(sinsqrt(3)-sinsqrt(2))+(cossqrt(3)-cos sqrt(2))]`
`=8sqrt(2)sin((sqrt(3)-sqrt(2))/(2))cos((pi)/(4)+(sqrt(2)+sqrt(3))/(2))`
As `sin((sqrt(3)-sqrt(2))/(2))gt0` and`(pi)/(2)lt(sqrt(2)+sqrt(3))/(2)ltpi`
`therefore D_(1)lt0`
Hence `f(x)gt0AA x in R`,