It is given that for the function `f` given by `f(x)=x^3+b x^2+a x` , `x in [1,\ 3]` . Rolles theorem holds with `c=2+1/(sqrt(3))` . Find the values of `a` and `b` .

Explanation

Given, `f(x)=x^{3}+b x^{2}+ax`


Rolle's theorem states that if `f(x)` be continuous on `[a, b]`, differentiable on `(a, b)` and `f(a)=f(b) `then there exists some `c` between `a` and `b` such that `f(c)=0`

Given` (a, b)=(1,3)` and `c=2+\frac{1}{\sqrt{3}}`

`f(1)=1+a+b` and

`f(3)=27+9b+3a`

According to Rolle's Theorem, `f(1)=f(3)`

`\Rightarrow a+b+1=3 a+9 b+27 `

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