IIT-JEE 2008 Paper 1 Offline | Application of Derivatives Question 31 | Mathematics

1

IIT-JEE 2008 Paper 1 Offline

MCQ (Single Correct Answer)

+3

-1

The total number of local maxima and local minima of the function

$$f(x) = \left\{ {\matrix{ {{{(2 + x)}^3},} & { - 3 < x \le - 1} \cr {{x^{2/3}},} & { - 1 < x < 2} \cr } } \right.$$ is

A

0

B

1

C

2

D

3

2

IIT-JEE 2007

MCQ (Single Correct Answer)

+3

-0.75

The tangent to the curve $$y = {e^x}$$ drawn at the point $$\left( {c,{e^c}} \right)$$ intersects the line joining the points $$\left( {c - 1,{e^{c - 1}}} \right)$$ and $$\left( {c + 1,{e^{c + 1}}} \right)$$

A

on the left of $$x=c$$

B

on the right of $$x=c$$

C

at no point

D

at all points

3

IIT-JEE 2007

MCQ (Single Correct Answer)

+4

-1

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 line $$y=x$$ meets $$y = k{e^x}$$ for $$k \le 0$$ at

A

no point

B

one point

C

two points

D

more than two points

4

IIT-JEE 2007

MCQ (Single Correct Answer)

+4

-1

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

$${\log _e}2$$