gorgeousgen9487

2022-07-08

By using the Intermediate value theorem.

Show that f is continuos on $\left[-1,1\right]$, then there exist n in the natural numbers such that the equation $f\left(x\right)+n=n\left({e}^{x}\right)$ has a solution in $\left[-1,1\right]$.

I'm having trouble solving this problem. Any help is appreciated.

kawiarkahh

Expert

Define $g\left(x\right):=f\left(x\right)+n-n\left({e}^{x}\right)$. Then $g$ is continuous and we have $g\left(-1\right)=f\left(-1\right)+n-\frac{n}{e}$ and $g\left(1\right)=f\left(1\right)+n-ne$. Can you finish it from here?