Question

calculus help, from this graph, please indicate the intervals where the function is positive and the intervals where the derivative is negative?

Answer 1

a)

In general, we say that a function f(x) is positive on an interval (a,b) if

`f(x)>0,x\in(a,b)`

In other words, the positive parts of a function are those sections of it above the x-axis.

Therefore, in our case,

`function\text{ }positive:(-\infty,-1)\cup(0,1)\cup(2.5,\infty)`

The function is positive on (-inf, -1)U(0, 1)U(2.5, +inf)

b)

On the other hand, a function is decreasing on a certain interval if its derivative on such interval is less than zero.

Therefore, we need to identify the intervals at which the graph is decreasing by using the definition below

Notice that -0.6 is just an approximation because we cannot know the exact value of the leftmost minimum of the function due to the lack of accuracy of the grid.

Hence, the derivative of the function is negative on the interval (-inf, -0.6)U(0.5, 2)