Root?

Algebra Level 4

Let the function $f$ be defined by $f(x) = ke^x + x$ , where $k$ is a real number independent of $x$ .

The intermediate value theorem shows that this function has at least one root in the interval $]0,1[$ .

Which one of the following intervals can $k$ belong?

Clarification : $e$ denotes Euler's number , $e \approx 2.71828$ .

\left ]-e,-\frac { 1 }{ e } \right [

\left ] 0, \frac { 1 }{ e }\right [

\left ]-\frac { 1 }{ e } , 0\right [

\left ]\frac { 1 }{ e } ,1\right [

2 solutions

Joao Miguel Coelho
May 10, 2016

$f$ is continuous in $\mathbb{R}$ . Therefore, it is continuous in $[0,1]$ .

The intermediate value theorem grants the existence of a root in $]0,1[$ . Then:

$f\left( 0 \right) \times f\left( 1 \right) <0$ $\Leftrightarrow$ $(k{ e }^{ 0 }+0)(k{ e }^{ 1 }+1)<0$ $\Leftrightarrow$ $k(ke+1)<0$ $\Leftrightarrow$ $k\in ]-\frac { 1 }{ e } ,0[$

Gaurav Chahar
May 10, 2016

f(0).f(1) must be negative as in the given interval f is either decreasing or increasing so it must have opposite sign values at x=0 and x=1. We can check the monotonous behavior of f by checking it's first derivative's sign in the interval

Root?

2 solutions

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