CGPET 2015 Solved Paper

Let f(x)=

1

3

ax3+

1

2

bx2+cx
Then, f(x) is a polynomial.
So, it is continuous in R.
Now, f(0)=0
and f(1)=

a

3

+

b

2

+c=

2a+3b+6c

6

⇒f(1)=0[∵2a+3b+6c=0,given]
∵ f(x) is a polynomial, so it is differentiable in R, so in (0,1).
Hence, by Rolle's theorem, there exists atleast one point x∊(0,1), there exists such that
f′(x)=0
⇒ ax2+bx+c=0
Hence, required interval is (0,1).