## Assertions about monotonicity and extremes

Determine whether the following assertions are true.

• #### Variant 1

A function $$f$$ is increasing on the interval $$I$$ if and only if: $$\forall x,y\in I, x\ne y: \frac{f(x)-f(y)}{x-y}> 0$$

• #### Variant 2

If a function $$f$$ is non-decreasing on the interval $$(-\infty,a\rangle$$ and non-increasing on $$\langle a,\infty)$$ for some $$a\in\mathbb R$$, then $$f$$ has a maximum.

• #### Variant 3

If the function $$f$$ has a minimum at the point $$a\in\mathbb R$$, then there exists $$\varepsilon>0$$ such that $$f$$ is non-increasing on the interval $$(a-\varepsilon,a]$$ and non-decreasing on $$[ a,a+\varepsilon)$$.

• #### Variant 4

If the function $$f$$ has limit $$\infty$$ at point $$c$$, then the value of the function $$f$$ on every punctured neighborhood $$P_\delta(c)$$ is unlimited.

• #### Variant 5

If for every punctured neighborhood $$P_\delta(c)$$ the value of the function $$f$$ is not bounded above, then $$\displaystyle \lim_{x\to c}f(x)=\infty$$.