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Differentiation question

2015 · Shift 2 · Q38

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PYQ Library
JEE Advanced
Mathematics
Differentiation
2015 · Shift 2 · Q38

Differentiation question

2015 · Shift 2 · Q38

JEE AdvancedMathematicsDifferentiationMultiple correct+4 / −1

Let

F:R \to R

be a thrice differentiable function. Suppose that

F\left( 1 \right) = 0,F\left( 3 \right) = - 4

and

F'\left( x \right) \lt 0

for all

x \in \left( {{1 \over 2},3} \right).

Let

f\left( x \right) = xF\left( x \right)

for all

x \in R.

The correct statement(s) is (are)

A
$f'\left( 1 \right) \lt 0$
B
$f\left( 2 \right) \lt 0$
C
$f'\left( x \right) e 0$ for any $x \in \left( {1,3} \right)$
D
$f'\left( x \right) = 0$ for some $x \in \left( {1,3} \right)$

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