JEE Advanced 2022 Paper 2 · Mathematics · Sets, Relations and Functions · medium

Let f(x)=∫0^xt(t-1)(t-2) dt for x∈mathbb R.

One or more options correct

Let

f(x)=\displaystyle\int_{0}^{x}t(t-1)(t-2)\,dt

for

x\in\mathbb R

. Which of the following statements is/are correct?

Show answer and solution

Answer: A, B, C

Integrating,

f(x)=\frac{x^4}4-x^3+x^2=\frac{x^2(x-2)^2}{4}

. Option A is true:

f'(x)=x(x-1)(x-2)

changes sign from negative to positive at

0

, from positive to negative at

1

, and from negative to positive at

2

. Option B is true:

f(x)=0

only at

x=0

and

x=2

, both double roots. Option C is true:

\int_0^2\frac{x^2(x-2)^2}4dx=\frac14\cdot2^5B(3,3)=\frac14\cdot\frac{32}{30}=\frac4{15}

. Option D is false:

f(1)=\frac14-1+1=\frac14\ne\frac12

Original question written in the JEE Advanced 2022 Paper 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake