If `a, b and c` are the sides of a `triangle ABC,` then `a^(1//p) + b^(1//p) - c^(1//p),` where `pgt 1,` is

Options

A. always negative
B. always positive
C. always zero
D. positive if 1 `lt` p `lt` 2 and negative if p `gt` 2

Answer

B

Explanation

Consider any equilateral triangle :
c = b = a = 1 unit
Take value of p between 1 & 2 i.e., `(3)/(2)`
`therefore a^(.^(1)//_(p))+b^(.^(1)//_(p))-c^(.^(1)//_(p))=(1)^(.^(2)//_(3))+(1)^(.^(2)//_(3))-(1)^(.^(2)//_(3))`
`=1 + 1 - 1 = 1 gt 0`
Take value of p greater than 2 i.e., 3.
`therefore a^(.^(1)//_(p))+b^(.^(1)//_(p))-c^(.^(1)//_(p))=(1)^(.^(1)//_(3))+(1)^(.^(1)//_(3))-(1)^(.^(1)//_(3))=1 gt 0`.
`therefore` By cosidering all the options carefully, we came to a conclusion that option (b) is correct.