Let `f(x)=(x-1)(x-2)(x-3)` on the interval [1, 3]. Prove that there is more than one c in (1, 3) such that f'( c) = 0.

Verify Mean Value theorem, if f(x)=x^(3)-5x^(2)-3x in the interval [a, b], where a = 1 and b = 3. Find all cin(1,3) for which f'( c) = 0.

Let f(x)=4x^3-3x^2-2x+1, use Rolles theorem to prove that there exist c ,\ 0

Let f(x)=((x-3)(x+2)(x+5))/((x+1)(x-7)) . Find the intervals were f(x) is positive or negative .

Let f(x)=((x-3)(x+2)(x+6))/((x+1)(x-5)) Find intervals where f(x) is positive or negative

If mean value theorem holds good for the function f(x)=(x-1)/(x) on the interval [1,3] then the value of 'c' is 2 (b) (1)/(sqrt(3))( c) (2)/(sqrt(3))(d)sqrt(3)

Let f(x) and g(x) be the differentiable functions for 1<=x<=3 such that f(1)=2=g(1) and f(3)=10 .Let there exist exactly one real number c in(1,3) such that 3f'(c)=g'(c) ,then the value of g(3) must be

Let f(x)=Max .{x^(2),(1-x)^(2),2x(1-x)}wherexin[0,1] If Rolle's theorem is applicable for f(x) on largestpossible interval [a,b] then the value of 2(a+b+c) when c in[a,b] such that f(c)=0, is

Using Lagrange's theorem , find the value of c for the following functions : (i) x^(3) - 3x^(2) + 2x in the interval [0,1/2]. (ii) f(x) = 2x^(2) - 10x + 1 in the interval [2,7]. (iii) f(x) = (x-4) (x-6) in the interval [4,10]. (iv) f(x) = sqrt(x-1) in the interval [1,3]. (v) f(x) = 2x^(2) + 3x + 4 in the interval [1,2].

Let f(x)= x^4- 3x^3 + 2x^2-3x+ a .If f(sqrt-1)= 0 and a in R then prove that f (x) < 0 for x in (1,2) .