Manipal Entrance Test 2020 Math 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)=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).