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 exam

Monjur Alahi

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


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Salim Ahmed Kawsar

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(a) For `y= (x)/(k)` to be tangent to the curve `y=e^(x), k= 1//e`.
For `y=e^(x)` to meet `y=(x)/(k)` at two points, we should have `k lt (1)/(e)`
`" "rArr k in (0, (1)/(e))` as `k gt 0`.

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