Question

estimate the intervals where f is increasing or decreasing.
increasing:
decreasing:
estimate the point at which the graph of f has a local maximum or a local minimum.
local maximum: (
local minimum: (

Explanation:

Step1: Determine increasing intervals

A function \(y = f(x)\) is increasing when the slope of the tangent line is positive (i.e., as \(x\) increases, \(y\) increases). Looking at the graph, we can see that the function is increasing on the intervals \((-\infty,2)\) and \((5,\infty)\).

Step2: Determine decreasing intervals

A function \(y = f(x)\) is decreasing when the slope of the tangent line is negative (i.e., as \(x\) increases, \(y\) decreases). From the graph, the function is decreasing on the interval \((2,5)\).

Step3: Locate local maximum

A local maximum occurs at a point where the function changes from increasing to decreasing. At \(x = 2\), the \(y\)-value (by estimating from the graph) is \(44\). So the local maximum is \((2,44)\).

Step4: Locate local minimum

A local minimum occurs at a point where the function changes from decreasing to increasing. At \(x=5\), the \(y\)-value (by estimating from the graph) is \(-18\). So the local minimum is \((5, - 18)\).

Answer:

Increasing: \((-\infty,2)\cup(5,\infty)\)
Decreasing: \((2,5)\)
Local maximum: \((2,44)\)
Local minimum: \((5,-18)\)