Verify LMVT for the following function `f(x)=x+1/x,x in [1,3].`

Explanation

The function f given as `f(x)=x+(1)/(x)` is a polynomial function. Hence, it is continuous on [1, 3] and differentiable on (1, 3). Thus, the function f satisfies the conditions of LMVT.
`:.` there exists `c in (1, 3)` such that
`f'( c )=(f(3)-f(1))/(3-1)" "...(1)`
Now, `f(x)=x+(1)/(x)`
`:.f(1)=1+(1)/(1)=2" and "f(3)=3+(1)/(3)=(10)/(3)`
Also, `f'(x)=("d")/("dx")(x+(1)/(x))=1-(1)/(x^(2))`
`:.f'(c)=1-(1)/(c^(2))`
`:.` from (1), `1-(1)/(c^(2))=((10)/(3)-2)/(3-1)=((4//3))/(2)=(2)/(3)`
`:.(1)/(c^(2))=1-(2)/(3)=(1)/(3)`
`:.c^(2)=3`
`:.c=sqrt(3)in(1, 3)`
Hence, the LMVT is verified.