# Thread: Need to demonstrate that the slope is strictly positive

1. ## Need to demonstrate that the slope is strictly positive

If I have a multivariable equation whose slope is always positive, say $f(x,y,z)=x^2+y^2+z^2$ how do I demonstrate that the slope is always positive?

I imagine this involves partial derivatives but need some guidance.

Thanks

2. Originally Posted by rainer
If I have a multivariable equation whose slope is always positive, say $f(x,y,z)=x^2+y^2+z^2$ how do I demonstrate that the slope is always positive?

I imagine this involves partial derivatives but need some guidance.

Thanks

How do you define the slope of a multivariable function??

Tonio

3. Oh yeah, good point.

Let me give a few more parameters.

First, I reduce the equation to 3 variables:

$f(x,y)=x^2+y^2$

So it's a 3D graph. I am interested in the slope on the 2D x-y plane or "cross-section" of the origin.

Does that narrow it down enough?

4. I don't understand your definition. The definiton I am familiar with says a function $f:\mathbb{R}^n\to \mathbb{R}$ has positive slope iff for every injective curve $c=(c^1,\cdots,c^n)a,b)\to \mathbb{R}^n" alt="c=(c^1,\cdots,c^n)a,b)\to \mathbb{R}^n" /> whose coordinate functions $c^i$ are increasing, the derivative of $f \circ c:\mathbb{R}\to \mathbb{R}$ is positive. Is this what you want?

I don't understand your definition. The definiton I am familiar with says a function $f:\mathbb{R}^n\to \mathbb{R}$ has positive slope iff for every injective curve $c=(c^1,\cdots,c^n)a,b)\to \mathbb{R}^n" alt="c=(c^1,\cdots,c^n)a,b)\to \mathbb{R}^n" /> whose coordinate functions $c^i$ are increasing, the derivative of $f \circ c:\mathbb{R}\to \mathbb{R}$ is positive. Is this what you want?