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
$$\left( {0,1} \right)$$
2
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 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
3
IIT-JEE 2007
MCQ (Single Correct Answer)
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
4
IIT-JEE 2005 Screening
MCQ (Single Correct Answer)
If $$P(x)$$ is a polynomial of degree less than or equal to $$2$$ and $$S$$ is the set of all such polynomials so that $$P(0)=0$$, $$P(1)=1$$ and $$P'\left( x \right) > 0\,\,\forall x \in \left[ {0,1} \right],$$ then
