Verify Rolles theorem for the function `f(x)=(x-a)^m(x-b)^n` on the interval `[a , b],` where `m , n` are positive integers.

Explanation

rolles theorem state that any real valued differentiable function which gives equal value at two distinct point will have a point in between them where `d(f(x) )= 0`

`f(x) = (x-a)^m (x-b)^n ; [a,b] `

for `x=a => f(x=a) = (a-a)^m (a-b)^n= 0`

for `x=b => f(x=b) = (b-a)^m(b-b)^n = 0`

`:. f(a) = f(b) `

now, `d/(dx) (v(x)) = d/(dx) (g(x) h(x))= h(x) d(g(x)) + g(x)d(h(x)) `

`d(f(x)) = d/(dx) ((x-a)^m(x-b)^n) `

`= (x-b)^n m (x-a)^(m-1) + (x-a)^m n(x-b)^(n-1)`

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