<p>The <a href="https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions#Nonlinear_optimization_problem" rel="nofollow noreferrer">Karush-Kuhn-Tucker theorem</a> would work, because
<span class="math-container">$$
\frac{\partial f(\mathbb{x})}{\partial x_i} + \mathbb{\lambda} \frac{\partial \mathbb{g}(\mathbb{x})}{\partial x_i} = 0
$$</span>
implies
<span class="math-container">$$
\frac{\partial f(\mathbb{x})}{\partial x_i} \neq \infty,
$$</span>
since the budget, technology etc. constraints have finite slopes<sup>1</sup> and <span class="math-container">$\lambda$</span> is a real-valued vector.</p>
<hr />
<p><sup>1</sup> The 'maximization problem' is not defined in the question, but these constraints are usually linear.</p>

CC BY-SA 4.0
2024-07-16T05:27:46.013Z
2024-07-16T05:27:46.013Z
Giskard
<p>I'm a first year grad student and something we've been using to ensure a maximization problem doesn't admit <span class="math-container">$0 \in \mathbb R^d$</span> as a solution is that the derivative (gradient) of the production/utility function is infinite at 0. As intuitive as this is, I haven't found anything that proves that it actually works. Does anyone know any references that do it? Thanks in advance.</p>

CC BY-SA 4.0
2024-07-16T01:12:57.870Z
See [Inada conditions](https://en.wikipedia.org/wiki/Inada_conditions)
Herr K.
<utility><economic-growth><profit-maximization>
Infinite derivative at zero bundle
Arthur
# Infinite derivative at zero bundle

(Asked by: username_0 on 2024-07-15)

I'm a first year grad student and something we've been using to ensure a maximization problem doesn't admit $0 \in \mathbb R^d$ as a solution is that the derivative (gradient) of the production/utility function is infinite at 0. As intuitive as this is, I haven't found anything that proves that it actually works. Does anyone know any references that do it? Thanks in advance.

**Tags:** utility, economic-growth, profit-maximization

> **Comments:**
> **username_1**: See [Inada conditions](https://en.wikipedia.org/wiki/Inada_conditions)  

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## 1 answers

### Answer by username_2 on 2024-07-16. Score: 0 | Accepted answer

The [Karush-Kuhn-Tucker theorem](https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions#Nonlinear_optimization_problem) would work, because
$$
\frac{\partial f(\mathbb{x})}{\partial x\_i} + \mathbb{\lambda} \frac{\partial \mathbb{g}(\mathbb{x})}{\partial x\_i} = 0
$$
implies
$$
\frac{\partial f(\mathbb{x})}{\partial x\_i} \neq \infty,
$$
since the budget, technology etc. constraints have finite slopes1 and $\lambda$ is a real-valued vector.

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1 The 'maximization problem' is not defined in the question, but these constraints are usually linear.


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