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1

IIT-JEE 2006

MCQ (More than One Correct Answer)
Let $$f\left( x \right) = \left\{ {\matrix{ {{e^x},} & {0 \le x \le 1} \cr {2 - {e^{x - 1}},} & {1 < x \le 2} \cr {x - e,} & {2 < x \le 3} \cr } } \right.$$ and $$g\left( x \right) = \int\limits_0^x {f\left( t \right)dt,x \in \left[ {1,3} \right]} $$
then $$g(x)$$ has
A
local maxima at $$x=1+In$$ $$2$$ and local minima at $$x=e$$
B
local maxima at $$x=1$$ and local minima at $$x=2$$
C
no local maxima
D
no local minima
2

IIT-JEE 2006

MCQ (More than One Correct Answer)
$$f(x)$$ is cubic polynomial with $$f(2)=18$$ and $$f(1)=-1$$. Also $$f(x)$$ has local maxima at $$x=-1$$ and $$f'(x)$$ has local minima at $$x=0$$, then
A
the distance between $$(-1,2)$$ and (a$$f(a)$$) where $$x=a$$ is the point of local minima is $$2\sqrt 5 $$
B
$$f(x)$$ is increasing for $$x \in \left[ {1,2\sqrt 5 } \right]$$
C
$$f(x)$$ has local minima at $$x=1$$
D
the value of $$f(0)=15$$
3

IIT-JEE 1999

MCQ (More than One Correct Answer)
The function $$f\left( x \right) = \int\limits_{ - 1}^x {t\left( {{e^t} - 1} \right)\left( {t - 1} \right){{\left( {t - 2} \right)}^3}\,\,\,{{\left( {t - 3} \right)}^5}} $$ $$dt$$ has a local minimum at $$x=$$
A
$$0$$
B
$$1$$
C
$$2$$
D
$$3$$
4

IIT-JEE 1998

MCQ (More than One Correct Answer)
Let $$h\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \right)^3}$$ for every real number $$x$$. Then
A
$$h$$ is increasing whenever $$f$$ is increasing
B
$$h$$ is increasing whenever $$f$$ is decreasing
C
$$h$$ is decreasing whenever $$f$$ is decreasing
D
nothing can be said in general.

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