Haley Madden

2022-07-23

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

Tolamaes04

Beginner2022-07-24Added 12 answers

${f}^{\prime}(x)=k(x+{e}^{x}{)}^{k-1}\times (1+{e}^{x})=0$

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.

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.

Zoagliaj

Beginner2022-07-25Added 2 answers

At critical points ${f}^{\prime}(x)=0$ so

$0=k(x+{e}^{x}{)}^{k-1}\times (1+{e}^{x})$

$k$ and $(1+{e}^{x})$ are positive so $0=x+{e}^{x}$

${x}_{n}=-{e}^{{x}_{n+1}}$ so $x\approx -0.56714329$

$0=k(x+{e}^{x}{)}^{k-1}\times (1+{e}^{x})$

$k$ and $(1+{e}^{x})$ are positive so $0=x+{e}^{x}$

${x}_{n}=-{e}^{{x}_{n+1}}$ so $x\approx -0.56714329$

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