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If a continuous function f defined on th...

If a continuous function `f` defined on the real line R assume 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(x)= ke^(x)-x`, for all real x where k is a real constant.
The line `y=x` meets `y=ke^(x)` for `k le 0` at

A

no point

B

one point

C

two points

D

more than two points

Text Solution

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The correct Answer is:
B
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Knowledge Check

  • If a continous founction of 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 values is negative, then the equation f(x)=0 has a root in R. Considetr f(x)=ke^(x)-x for all real x where k is real constant. The positive value of k for which ke^(x)-x=0 has only root is

    A
    `1/e`
    B
    1
    C
    e
    D
    `log_(e)2`

  • If a continous founction of 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 values is negative, then the equation f(x)=0 has a root in R. Considetr f(x)=ke^(x)-x for all real x where k is real constant. For k gt 0, the set of all values of k for which ke^(x)-x=0 has two distinct, roots, is

    A
    `(0,(1)/(e))`
    B
    `((1)/(e),1)`
    C
    `((1)/(e),oo)`
    D
    `(0,1)`

  • If the equation 4x^(3)+5x+k=0(k in R) has a negative real root then

    A
    k=0
    B
    `-ooltklt0`
    C
    `0ltkltoo`
    D
    `-ooltkltoo`

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