# Need to demonstrate that the slope is strictly positive

• Feb 21st 2010, 08:37 AM
rainer
Need to demonstrate that the slope is strictly positive
If I have a multivariable equation whose slope is always positive, say $\displaystyle f(x,y,z)=x^2+y^2+z^2$ how do I demonstrate that the slope is always positive?

I imagine this involves partial derivatives but need some guidance.

Thanks
• Feb 21st 2010, 11:10 AM
tonio
Quote:

Originally Posted by rainer
If I have a multivariable equation whose slope is always positive, say $\displaystyle f(x,y,z)=x^2+y^2+z^2$ 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
• Feb 22nd 2010, 09:41 AM
rainer
Oh yeah, good point.

Let me give a few more parameters.

First, I reduce the equation to 3 variables:

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

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?
• Feb 22nd 2010, 01:12 PM
I don't understand your definition. The definiton I am familiar with says a function $\displaystyle f:\mathbb{R}^n\to \mathbb{R}$ has positive slope iff for every injective curve $\displaystyle c=(c^1,\cdots,c^n):(a,b)\to \mathbb{R}^n$ whose coordinate functions $\displaystyle c^i$ are increasing, the derivative of $\displaystyle f \circ c:\mathbb{R}\to \mathbb{R}$ is positive. Is this what you want?
• Feb 24th 2010, 06:57 AM
rainer
Quote:

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