Verify Rolle theorem for the function `f(x)=log{(x^2+a b)/(x(a+b))}on[a , b],` where `0

Explanation

Here, `f(x) = log{(x^2+ab)/(x(a+b))}`

We have to find a point `c in (a,b)`such that ,

`f'(c) = 0`

Now, `f'(x) = (x(a+b))/((x^2+ab))((x(a+b)(2x) - (x^2+ab)(a+b))/(x(a+b))^2) `

`= (2x^2(a+b) - x^2(a+b) - ab(a+b))/((x^2+ab)(x(a+b))`

`= (x^2(a+b) - ab(a+b))/((x^2+ab)(x(a+b))`

`=>f'(x)=1/(x^2+ab)[x-(ab)/x]`

Now, for any value, `c`, `f'(c) = 0`.

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