IIT-JEE 2012 Paper 2 Offline | Application of Derivatives Question 21 | Mathematics

IIT-JEE 2012 Paper 2 Offline

MCQ (More than One Correct Answer)

-1

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

$$f$$ has a local maximum at $$x=2$$

$$f$$ is decreasing on $$(2, 3)$$

there exists some $$c \in \left( {0,\infty } \right),$$ such that $$f'(c)=0$$

$$f$$ has a local minimum at $$x=3$$

IIT-JEE 2009 Paper 2 Offline

MCQ (More than One Correct Answer)

-2

For the function $$$f\left( x \right) = x\cos \,{1 \over x},x \ge 1,$$$

for at least one $$x$$ in the interval $$\left[ {1,\infty } \right)$$, $$f\left( {x + 2} \right) - f\left( x \right) < 2$$

$$\mathop {\lim }\limits_{x \to \infty } f'\left( x \right) = 1$$

for all $$x$$ in the interval $$\left[ {1,\infty } \right)f\left( {x + 2} \right) - f\left( x \right) > 2$$

$$f'(x)$$ is strictly decreasing in the interval $$\left[ {1,\infty } \right)$$

IIT-JEE 2006

MCQ (More than One Correct Answer)

-1.25

$$f(x)$$ 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

the distance between $$(-1,2)$$ and (a$$f(a)$$) where $$x=a$$ is the point of local minima is $$2\sqrt 5 $$

$$f(x)$$ is increasing for $$x \in \left[ {1,2\sqrt 5 } \right]$$

$$f(x)$$ has local minima at $$x=1$$

the value of $$f(0)=15$$

IIT-JEE 2006

MCQ (More than One Correct Answer)

-1.25

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

local maxima at $$x=1+In$$ $$2$$ and local minima at $$x=e$$

local maxima at $$x=1$$ and local minima at $$x=2$$

no local maxima

no local minima

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