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

Answer:

Let's assume from the graph (since no specific values are given in text but we can conceptually analyze) if the function has local minima and maxima points. A function $y = f(x)$ is increasing when the slope of the tangent line to the curve is positive. If we assume the local minimum occurs at $x = a$ and local maximum occurs at $x = b$ (where $a < b$) from observing the graph's general shape, the function is increasing on the intervals where the graph is going up - hill. Typically for a cubic function with one local minimum and one local maximum, it is increasing on $(-\infty,a)\cup(b,\infty)$. Without seeing the exact graph values, we can't give numerical intervals. But if we had values say local min at $x = - 2$ and local max at $x=3$, the answer in interval notation would be $(-\infty,-2)\cup(3,\infty)$.

Explanation:

Step1: Recall increasing - function property

A function $y = f(x)$ is increasing when $f'(x)>0$. Geometrically, the graph of the function goes uphill as we move from left - to - right.

Step2: Identify critical points

Critical points are where $f'(x) = 0$ or $f'(x)$ is undefined. For a cubic polynomial, these are the local minima and maxima points.

Step3: Determine increasing intervals

The function is increasing in the intervals between and outside the critical points where the graph has a positive slope.