Verify Lagrange's mean value theorem for the function `f(x)=(x-3)(x-6)(x-9)` on the interval `[3. 5]`

Explanation

Given, `f(x)=(x-3)(x-6)(x-9)` is a polynomial function which is
continuous and differentiable everywhere and in `[3,5]`


`f(3)=0*(-3)*(-6)=0`


`f(5)=2*(-1)*(-4)=8`


All condition of Langrage's theorem is satisfied and so there exist `c in (1,4)` such that `f'(c)=(f(b)-f(a))/(b-a)`


`f(x)=(x-3)(x-6)(x-9)`


`f'(x)=(x-6)(x-9)+(x-3)(x-9)+(x-3)(x-6)`


`f'(c)=(c-6)(c-9)+(c-3)(c-9)+(c-3)(c-6)`


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