consider the cubic polynomial function f graphed here. where is f increasing?

Answer:

The function (f) is increasing on the intervals where the graph has a positive - slope. Without the actual graph details (since the provided image link doesn't work), in general, we look for the parts of the graph where as (x) increases, (y) also increases. If we assume we can identify the critical points (where the derivative is zero or undefined) of the cubic function from the graph as (x = a) and (x = b) ((a\lt b)), then the function is increasing on the intervals ((-\infty,a)) and ((b,\infty)) (this is a general form based on the behavior of cubic functions).

Explanation:

Step1: Recall increasing - function property

A function (y = f(x)) is increasing when (f^{\prime}(x)>0). For a cubic function (y=ax^{3}+bx^{2}+cx + d), its derivative (y^{\prime}=3ax^{2}+2bx + c).

Step2: Identify critical points

Critical points of (y = f(x)) are found by setting (f^{\prime}(x)=0). For a cubic function, (f^{\prime}(x)) is a quadratic function. The roots of (f^{\prime}(x)) divide the (x) - axis into intervals.

Step3: Determine sign of derivative

We test the sign of (f^{\prime}(x)) in the intervals separated by the critical points. If (f^{\prime}(x)>0) in an interval, then (f(x)) is increasing in that interval. On a graph, this corresponds to the parts of the curve that are rising as we move from left - to - right.