JEE Advanced 2015 · Paper 2 · Mathematics Q59

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MathematicsMultiple correct+4 / -2Question 59 of 60

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Let $F:\mathbb R\to\mathbb R$ be a thrice differentiable function. Suppose that $F(1)=0$ , $F(3)=-4$ and $F'(x)<0$ for all $x\in(1/2,3)$ . Let $f(x)=xF(x)$ for all $x\in\mathbb R$ .

The correct statement(s) is(are)

f'(1)<0

f(2)<0

f'(x)\ne0

for any

x\in(1,3)

f'(x)=0

for some

x\in(1,3)

Answer and worked solution

Official answer: A, B, C

Derivative. $f'=F+xF'$ .

Sign of F. $F$ decreases on $(\frac12,3)$ from $F(1)=0$ , so $F<0$ on $(1,3]$ .

(A) $f'(1)=F(1)+F'(1)=F'(1)<0$ . True.
(B) $f(2)=2F(2)<0$ . True.
(C) On $(1,3)$ , $F<0$ and $F'<0$ , so $f'<0$ and never vanishes. True.
(D) False.

(A), (B), (C)

Topics:Monotonicity Derivatives of standard functions

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JEE Advanced 2015 · Paper 2, Mathematics question 59