Discuss the applicability of Rolles theorem on the function `f(x)={x^2+1,\ \ \ \ w h e n\ 0lt=xlt=1 3-x ,\ \ \ \ w h e n\ 1

Answer

N/a

Explanation

We have, `f(x) = {{:(x^(2)+1,if0lexle1),(3-x,if1lexle2):}`
We know that, polynomical function is everywhere continuous and differentiability.
So, `f(x)` is continuous and differentiable at all points except possibly at `x = 1`.
Now, check the differentiability at `x = 1`.
At `x = t`,
`LDH = underset(xrarr1^(-))(lim)(f(x)-f(1))/(x-1)`
` = underset(xrarr1)(lim) ((x^(2)+1)-(1+1))/(x-1)` `[:' f(x) =x^(2)+1, AA 0 le x le 1]`
`= underset(xrarr1)(lim)(x^(2)-1)/(x-1) = underset(xrarr1)(lim) ((x+1)(x-1))/(x-1)`
and `RDH = underset(xrarr1^(+))(lim)(f(x)-f(1))/(x-1)=underset(xrarr1)(lim)((3-x)f(1+1))/((x+1))`
`= underset(xrarr1)(lim)(3-x-2)/(x-1)= underset(xrarr1)lim(3-x-2)/(x-1)=underset(rarr1)(lim)(-(x-1))/(x-1)=-1`
`:. LHD ne RHD`
So, f(x) is not differentiable at `x = 1`.
Hence, polle's theorem is not applicable on the interval `[0,2]`.