# Why $\frac{d}{dt}f(x+t(y−x))<0$ if $x < y, f(y) < f(x)$

Here excerpt from a book:

Аssume that $$f$$ satisfies $$\nabla f(x) \ge 0$$ for all $$x$$, but is not nondecreasing, i.e., there exist $$x,y$$ with $$x < y$$ and $$f(y) < f(x)$$. By differentiability of $$f$$ there exists at $$t\in[0,1]$$ with

$$\frac{d}{dt}f\left(x+t(y−x)\right) =\nabla f\left(x+t\left(y−x\right)\right)^T(y−x)<0.$$

I don't understand why the derivative is less than zero? The function could look like on the image.

The book is Convex Optimization by Stephen Boyd and Lieven Vandenberghe, page 109.

• It's not that it's always negative, it's that its negative somewhere. – πr8 Jun 30 '16 at 15:41
• It isn't saying for all but rather for some. – Cameron Williams Jun 30 '16 at 15:41
• Suppose otherwise that $\frac{d}{dt}f(x+t(y-x))\geq 0$ for all $t\in[0,1]$. Can you find a contradiction? – yurnero Jun 30 '16 at 16:48

Let $g(t)\equiv f(x+t(y-x))$ be defined on $[0,1]$. Then, $g$ is continuous on $[0,1]$ and differentiable on $(0,1)$. So by the Mean Value Theorem, there is some $t^*\in(0,1)$ such that $$g'(t^*)=\frac{g(1)-g(0)}{1-0}=f(y)-f(x)<0.$$ It remains to expand the leftmost expression above.