Good Day! I,m aware of the basic concept of mean value theorem but the application of it in proving makes me confuse, this is how it goes:
By mean Value theorem: $$2 - t^{n-1} (1+t) = (1 - t)[θ^{n – 1} + (n - 1) θ^{n – 2} (1 + θ)$$ $$ < (2n - 1)(1 + t)$$ where $$t < θ < 1$$
How does it happen? I have no idea.... I think this is complex form of mean value theorem...Any idea would be a great help thanks!
it looks like you are applying the mean value theorem to the function $$ f(x) = x^{n-1}(1+x)=x^{n-1}+x^n, \, f'(x) = (n-1)x^{n-2}+nx^{n-1} $$ on the closed interval $[t, 1].$
we get $$f(1) - f(t)=f'(\theta) \text{ for some } \theta \in (t, 1.)$$
that is $$ 2- t^{n-1}(1+t) = (1-t)\left((n-1)\theta^{n-2}+n\theta^{n-1}\right)$$