# Determine the total number of critical points of the function f(x)=(x+e^x)^k, where k>0 is an integer

Determine the total number of critical points of the function $f\left(x\right)=\left(x+{e}^{x}{\right)}^{k}$, where $k>0$ is an integer
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Tolamaes04
${f}^{\prime }\left(x\right)=k\left(x+{e}^{x}{\right)}^{k-1}×\left(1+{e}^{x}\right)=0$
has only one solution which is where $x+{e}^{x}=0$ and that is the point that you want to approximate.
The answer should be negative so $x=0.567$ is problematic.
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Zoagliaj
At critical points ${f}^{\prime }\left(x\right)=0$ so
$0=k\left(x+{e}^{x}{\right)}^{k-1}×\left(1+{e}^{x}\right)$
$k$ and $\left(1+{e}^{x}\right)$ are positive so $0=x+{e}^{x}$
${x}_{n}=-{e}^{{x}_{n+1}}$ so $x\approx -0.56714329$