Question

Determine the interval(s) for which the function
shown below is decreasing.

Answer 1

Check the picture below.

Answer 2

The function in the graph is decreasing over the intervals
`$\llap{-}2.7$` to
`$\llap{-}1.2$` and
`$0.9$` to
`$1.8$`.

We can find the intervals where the function is increasing/decreasing by looking for the intervals where its derivative is positive/negative. The derivative of the function is shown in blue below the graph of the function.

A function can only change its direction from increasing to decreasing and vice versa between its critical points and the points where the function itself is undefined.

The critical points of the function are the points where the derivative is zero. In the graph, the derivative is zero at
`$x \approx -2.7$`,
`$x \approx -1.2$`,
`$x \approx 0.9$`, and
`$x \approx 1.8$`.

We can now look at the intervals between the critical points and the points where the function is undefined to see if the derivative is positive or negative on that interval. On the interval from
`$x = -3$` to
`$x \approx -2.7$`, the derivative is negative. This means that the function is decreasing on this interval.

On the interval from
`$x \approx -2.7$` to
`$x \approx -1.2$`, the derivative is positive. This means that the function is increasing on this interval.

On the interval from
`$x \approx -1.2$` to
`$x = 0$`, the derivative is negative. This means that the function is decreasing on this interval.

On the interval from x = 0 to
`$x \approx 0.9$`, the derivative is positive. This means that the function is increasing on this interval.

On the interval from
`$x \approx 0.9$` to
`$x \approx 1.8$`, the derivative is negative. This means that the function is decreasing on this interval.

And finally, on the interval from
`$x \approx 1.8$` to x = 2, the derivative is positive. This means that the function is increasing on this interval.

In conclusion, the function is decreasing over the intervals
`$\llap{-}2.7$` to
`$\llap{-}1.2$` and 0.9 to 1.8.