The total number of local maxima and local minimum of the
function $$f\left( x \right) = \left\{ {\matrix{
{{{\left( {2 + x} \right)}^3},} & { - 3 < x \le - 1} \cr
{{x^{2/3}},} & { - 1 < x < 2} \cr
} } \right.$$ is
A
$$0$$
B
$$1$$
C
$$2$$
D
$$3$$
2
IIT-JEE 2008
MCQ (Single Correct Answer)
Consider the two curves $${C_1}:{y^2} = 4x,\,{C_2}:{x^2} + {y^2} - 6x + 1 = 0$$. Then,
A
$${C_1}$$ and $${C_2}$$ touch each other only at one point.
B
$${C_1}$$ and $${C_2}$$ touch each other exactly at two points
C
$${C_1}$$ and $${C_2}$$ intersect (but do not touch ) at exactly two points
D
$${C_1}$$ and $${C_2}$$ neither intersect nor touch each other
3
IIT-JEE 2007
MCQ (Single Correct Answer)
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root in $$R$$. For example, if it is known that a continuous function $$f$$ on $$R$$ is positive at some point and its minimum value is negative then the equation $$f(x)=0$$ has a root in $$R$$.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.
For $$k>0$$, the set of all values of $$k$$ for which $$k{e^x} - x = 0$$ has two distinct roots is
A
$$\left( {0,{1 \over e}} \right)$$
B
$$\left( {{1 \over e},1} \right)$$
C
$$\left( {{1 \over e},\infty } \right)$$
D
$$\left( {0,1} \right)$$
4
IIT-JEE 2007
MCQ (Single Correct Answer)
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root in $$R$$. For example, if it is known that a continuous function $$f$$ on $$R$$ is positive at some point and its minimum value is negative then the equation $$f(x)=0$$ has a root in $$R$$.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.
The positive value of $$k$$ for which $$k{e^x} - x = 0$$ has only one root is
A
$${1 \over e}$$
B
$$1$$
C
$$e$$
D
$${\log _e}2$$
Questions Asked from Application of Derivatives
On those following papers in MCQ (Single Correct Answer)
Number in Brackets after Paper Indicates No. of Questions
