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Show that the eigenvalues of the Hessian of $$f(x_1,x_2) := x_1^2+x_1x_2+x_2^2+\ln (1+2e^{x_2})$$ are bounded, i.e., $$1 \leq \lambda_{\min} \left(\nabla^2 f(x)\right) \leq \lambda_{\max}\left(\nabla^2 f(x)\right) \leq 4$$

What I tried is finding gradient and then Hessian

$$ \nabla f(x) = \begin{bmatrix} 2x_1+x_2 \\ x_1+2x_2+\frac{2e^{x_2}}{1+2e^{x_2}} \end{bmatrix} $$

and Hessian is:

$$ \nabla^2 f(x) = \begin{bmatrix} 2 & 1 \\ 1 & 2 +\frac{2e^{x_2}}{(1+2e^{x_2})^2} \end{bmatrix} $$

Let $y=\frac{2e^{x_2}}{(1+2e^{x_2})^2}$. Then

$$ \nabla^2 f(x) = \begin{bmatrix} 2 & 1 \\ 1 & 2 +y \end{bmatrix} $$

Then,

$$ \lambda_{1,2} = \frac{4+y\pm \sqrt{y^2+4}}{2} $$

My questions are:

  1. Am I mistaken?

  2. Is this function a well-known function?

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  • $\begingroup$ The Hessian seems to depend on $x_2$. Must the boundedness of the eigenvalues hold for all $x_2 \in \mathbb R$? $\endgroup$ – Rodrigo de Azevedo May 12 at 17:22
  • $\begingroup$ First of all, show that $y$ is bounded! $\endgroup$ – Fakemistake May 12 at 17:29
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Hint: Show that $y$ is bounded for all $x_2\in\mathbb{R}$:

Part 1:

\begin{align}y=\frac{2e^{x_2}}{(1+2e^{x_2})^2}< \frac{1+2e^{x_2}}{(1+2e^{x_2})^2}=\frac{1}{1+2e^{x_2}}<1\end{align}

Part 2: Observe that \begin{align} \frac{2e^{x_2}}{(1+2e^{x_2})^2}=\frac{-1+1+2e^{x_2}}{(1+2e^{x_2})^2}=-\frac{1}{(1+2e^{x_2})^2}+\frac{1}{1+2e^{x_2}}\end{align} but since $(1+2e^{x_2})^2>1+2e^{x_2}$ we have $$\frac{1}{1+2e^{x_2}}>\frac{1}{(1+2e^{x_2})^2}$$ and thus $$-\frac{1}{(1+2e^{x_2})^2}+\frac{1}{1+2e^{x_2}}>0$$ Consequently

\begin{align}y=\frac{2e^{x_2}}{(1+2e^{x_2})^2}=-\frac{1}{(1+2e^{x_2})^2}+\frac{1}{1+2e^{x_2}}>0 \end{align}

Now you can do the final part with $0<y<1$

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  • $\begingroup$ For the part two you have an extra exponent for the first denominator. My question: how would you conclude the result using $0<y<1$ and closed form formula of $\lambda_{1,2}$ ? $\endgroup$ – Saeed May 12 at 18:53
  • $\begingroup$ In other words why $1 \leq \lambda \leq 4$? $\endgroup$ – Saeed May 12 at 19:07
  • $\begingroup$ @Saeed I've edited my answer and deleted the wrong part. Maybe it helps a little bit. $\endgroup$ – Fakemistake May 19 at 8:52

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