MHB Prove Monotony of Function: $f$ Strictly Decreasing

[awsomeman]

Let $f$ be differentiable from $(-\inf,0)$ to $(0,\inf)$ and let $f'(x)<0$ for all real numbers except 0 and $f'(0)=0$. Prove that f is strictly decreasing.

 

[Joppy]

You might want to begin by stating the definition of a decreasing function. Then consider some examples.

 

[HOI]

I would use "proof by contradiction". Suppose f is NOT strictly decreasing. Then there exist a, b, b> a, such that f(b)\ge f(a). So $f(b)- f(a)\ge 0$. Since b> a, b- a> 0 so $\frac{f(b)- f(a)}{b- a}\ge 0$. Now use the "mean value" property.