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Question

If $$f(x) = (x - a)^m (x - b)^n m, n$$ are positive integers, satisfies the conditions of Rolle's theorem on $$[a, b]$$, then find '$$c$$'.


Solution

$$f(x)=(x-a)^{m}(x-s)^{n}$$
$$f(a)=0$$
$$a(b)=0$$
thus by Rolle's theorem $$f\ C$$ st.
$$f'(c)=0$$                   $$c\in(a,b)$$
$$f'(x)=\dfrac{d}{dx}(f(x))=m(x-a)^{m-1}(x-b)^{n}+n(x-a)^{m}(x-b)^{n-1}$$
$$f'(x)=0$$
$$\Rightarrow m.(x-a)^{m-1}(x-b)^{n}+n(x-a)^{m}(x-b)^{n-1}=0$$
$$\Rightarrow (x-a)^{m-1}.(x-b)^{n-1}(m(x-b)+n(x-a))=0$$
$$\Rightarrow m(x-b)+n(x-a)=0$$
$$\Rightarrow x=\dfrac{mb+na}{m+n}$$
$$a<\dfrac{mb+na}{m+n}<b$$
$$\Rightarrow c=\dfrac{mb+na}{m+n}$$


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