Question

prove that 2x^3-3x^2-x+1 has atleast one root between 0 and 3​

Answer 1

Answer:

Let f

′

(x)=3ax

2

+2bx+c

⟹f(x)=ax

3

+bx

2

+cx+d (on integration)

Rolle's theorem states that if f(x) be continuous on [a,b], differentiable on (a,b) and f(a)=f(b) then there exists some c between a and b such that f

′

(c)=0

let (a,b)=(0,1)

f(0)=d and f(1)=a+b+c+d

But, f(0)=f(1).

⟹d=a+b+c+d

⟹a+b+c=0