# Building a Function with saddle point that has Certain Properties

I'm working on a problem and I'd like to compute a couple examples. For one example I'd like to create a function from $$\mathbb{R}^2 \rightarrow \mathbb{R}$$ with the following properties:

1. It has a saddle at zero
2. It goes to positive infinity on the x AND y axis (so maybe cross sections along these axes are upward opening parabolas)
3. It goes to negative infinity along the line $$x = y$$ AND $$x = -y$$ (so maybe cross sections along these axes are downward opening parabolas).

What is an example of a function with such properties?

Thanks!

• A degree $2$ polynomial obviously won't do. Have you tried a degree $4$ polynomial? Dec 20, 2020 at 20:11
• Haha, nope. I'm totally lost on what should I guess. Dec 20, 2020 at 20:14
• Here's a hint: Have you played with a monkey saddle? It only has three directions of interest, not four, but understanding this might give you an idea. Dec 20, 2020 at 20:19

For my particular problem: $$f(x,y) = x^4 - 6x^2y^2 +y^4$$