`f(x) = 1/(4x-1)` in `[1,4]`

Explanation

`f(x) = (1)/(4x-1)`
(i) The function f(x) has definite and unique value for each point in the interval [1,4].
`therefore` f(x) is continuous in [1,4] .
(ii) `f'(x) = (-4)/((4x-1)^(2))`
Which is defined in ]1, 4[.
`therefore` f(x) is differentiable in ]1, 4[.
Thus the conditions of Lagrange's Mean Value theorem satisfies.
Now there exits atleast one value of `c in ]1,4[` such that .
`f'(c) = (f(4) - f(1))/(4-1)`
`implies " " (-4)/((4c-1)^(2)) = ((1)/(15) - (1)/(3))/(3) = (-4)/(3 xx 15)`
`implies " " (4c-1)^(2) = 45`
`implies " " 4c-1= pm 3sqrt5`
`implies " " c = (1 pm 3sqrt5)/(4)`
`therefore " " c = (1 + 3sqrt5)/(4) in ]1, 4[`
Hence Lagrange's Mean Value theorem verified .