# Demetri is studying a new function g and has concluded that g is decreasing at x = 5. Which of the following can Demetri now conclude? [Choose all that apply] (Choices below)

## I say that it is II, V, and either III or IV? Can you just double-check? I. The slope of the tangent line to g at $x = 5$ is positive II. The slope of the tangent line to g at $x = 5$ is negative III. The rate of change of $g$ at $x = 5$ is positive IV. The rate of change of $g$ at $x = 5$ is negative V. $g ' \left(5\right) < 0$ VI. $g ' \left(5\right) > 0$

Mar 4, 2017

II, IV and V

#### Explanation:

The definition of an increasing or decreasing function is that

$\left\{\begin{matrix}y ' < 0 \\ y ' > 0\end{matrix}\right. \implies \left.\begin{matrix}y \text{ is decreasing" \\ y " is increasing}\end{matrix}\right.$

So if $g \left(5\right)$ is decreasing $\implies g ' \left(5\right) < 0$

$g ' \left(x\right)$ also represents the rate of change and the gradient of the tangent.

Hence the correct solutions are:

II, IV and V