IIT-JEE 2012 · Paper 2 · Mathematics Q56

MathematicsMultiple correct+4Question 56 of 60

If $f(x)=\displaystyle\int_0^x e^{t^2}(t-2)(t-3)\,dt$ for all $x\in(0,\infty)$ , then

f

has a local maximum at

x=2

f

is decreasing on

(2,3)

there exists some

c\in(0,\infty)

such that

f''(c)=0

f

has a local minimum at

x=3

Answer and worked solution

Official answer: A, B, C, D

Derivative. $f'(x)=e^{x^2}(x-2)(x-3)$ : positive on $(0,2)$ , negative on $(2,3)$ , positive on $(3,\infty)$ .

(A) $f'$ changes from + to − at 2: local maximum. True.
(B) $f'<0$ on $(2,3)$ . True.
(C) $f'(2)=f'(3)=0$ , so by Rolle's theorem $f''(c)=0$ for some $c\in(2,3)$ . True.
(D) $f'$ changes from − to + at 3: local minimum. True.

(A), (B), (C), (D)

IIT-JEE 2012 · Paper 2, Mathematics question 56