If `f(x)={(x^(alpha)logx , x > 0),(0, x=0):}` and Rolle's theorem is applicable to `f(x)` for `x in [0, 1]` then `alpha` may equal to (A) -2 (B) -1 (C) 0 (D) `1/2`

If f(x)={:{(2x+1,x le 0),(3(x+1),x gt 0):} . Find lim_(x to 0) f(x) and lim_(x to 1) f(x) .

Statement 1: If g(x) is a differentiable function, g(2)!=0,g(-2)!=0, and Rolles theorem is not applicable to f(x)=(x^2-4)/(g(x))in[-2,2],t h e ng(x) has at least one root in (-2,2)dot Statement 2: If f(a)=f(b),t h e ng(x) has at least one root in (-2,2)dot Statement 2: If f(a)=f(b), then Rolles theorem is applicable for x in (a , b)dot

Verify Rolle's theorem for the following f(x) = x^(3) -3x + 3 " in " 0 le x le 1

Explain why Rolle's theorem is not applicable to the following functions in the respective intervals. f(x)=|1/x|,x in[-1,1]

Explain why Rolle's theorem is not applicable to the following functions in the respective intervals. f(x)=tanx,x in[0,pi]

Let f(x)=(1+b^2)x^2+2b x+1 and let m(b) the minimum value of f(x) as b varies, the range of m(b) is (A) [0,1] (B) (0,1/2] (C) [1/2,1] (D) (0,1]

If f(x)={{:(x","" "xlt0),(1","" "x=0),(x^(2)","" "xgt0):}," then find " lim_(xto0) f(x)" if exists.

Let f(x ) = ( alpha x^2 )/(x+1),x ne -1 the value of alpha for which f(a) =a , (a ne 0) is

The value of c in Lagrange's mean value theorem for the function f(x) = x^(2) + 2x -1 in (0, 1) is

In [0,1] Lagranges Mean Value theorem in NOT applicable to f(x)={1/2-x ; x =1/2 b. f(x)={(sin x)/x ,x!=0 1,x=0^ c. f(x)=x|x| d. f(x)=|x|