If $f(x) = x^2 + kx + 1$ is a strictly increasing function on the interval $[1, 2]$,then what is the minimum value of $k$?

Consider the following statements $S$ and $R$:
$S$: Both $\sin x$ and $\cos x$ are decreasing functions in $\left( \frac{\pi}{2}, \pi \right)$.
$R$: If a differentiable function decreases in $(a, b)$,then its derivative also decreases in $(a, b)$.
Which of the following is true?

Find the intervals in which the function $f$ given by $f(x)=2 x^{2}-3 x$ is
$(a)$ increasing
$(b)$ decreasing

Find the intervals in which the function $f(x) = 6 - 9x - x^{2}$ is strictly increasing or strictly decreasing.

Observe the following statements:
$A: f(x) = 2x^3 - 9x^2 + 12x - 3$ is increasing outside the interval $(1, 2)$.
$R: f^{\prime}(x) < 0$ for $x \in (1, 2)$.
Then,which of the following is true?

Given $f(x) = \int_{-2}^{x} t \cdot g'(t) \, dt$ for $x \geq -2$,where $g$ is an increasing function,then: