if a+b+c=0 then the quadratic eqation 3x2+2bx+c has at least one root b/t (0,1)

Lets take f(x)=ax³ + bx² + cx + d

so, f' (x) = 3ax² + 2bx + c

Uding Rolles theorem, (Rolle's Theorem)   Let f(x) be continuous on [a, b], and differentiable on (a, b), and suppose that f (a) = f (b). Then there is some c with a < c < b such that f'(c) = 0

here (a , b) = (0 ,1)

f(0)= d and f(1) =a+b+c+d ; f(0) = f(1) which means a+b+c=0 which is true from given condition

hence there exists some c in btn (0, 1) where f '(c) =0

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This ques is solved using the ROLLE'S THEOREM

Let f '(x)= 3x2+2bx+c

=>  f(x)=x3+bx2+cx+k                where k is the integration constant

now f(0)=k    and f(1)=a+b+c+k =>          f(1)=0+k                   =>f(1)=k

So we get that f(0)=f(1)

  Using Rolle's Theorem we know that if f(a)=f(b)  then there exixsts at least one root c in (a,b)  such that f '(c)=0

 Now we  can say that 3x2+ 2bx+c  has at least one root   b/t (0,1) .