## Assertions about monotonicity and extremes

### Task number: 2974

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 \(\displaystyle \lim_{x\to c}f(x)=\infty \).