If `a, bgt0,a+b=1`, then the least value of
`(1+(1)/(a))(1+(1)/(b))`, is

Options

A. 3
B. 6
C. 9
D. 12

Answer

C

Explanation

We have,
`(1+(1)/(a))(1+(1)/(b))=((a+1)(b+1))/(ab)=(1+(a+b)+ab)/(ab)=(2+ab)/(ab)`
`implies" "(1+(1)/(a))(1+(1)/(b))=1+(2)/(ab)`
Now, `A.M. geG.M.`
`implies" "(a+b)/(2)gesqrt(ab)`
`implies" "(1)/(2)gesqrt(ab)implies(1)/(ab)ge4implies1+(2)/(ab)ge9implies(1+(1)/(a))(1+(1)/(b))ge9`
Hence, the least value of `(1+(1)/(a))(1+(1)/(b))` is 9.