Question

1. (8 points) given the function shown in the graph below, identify the intervals on which the function appears to be increasing or decreasing.

• increasing:
• decreasing:
• x values where f(x)=0:
• critical points (x, f(x))=

Explanation:

Step1: Recall increasing - decreasing rules

A function $y = f(x)$ is increasing when $f^{\prime}(x)>0$ (graph goes up from left - to - right) and decreasing when $f^{\prime}(x)<0$ (graph goes down from left - to - right). Critical points occur where $f^{\prime}(x)=0$ or $f^{\prime}(x)$ is undefined.

Step2: Identify increasing intervals

By observing the graph, the function is increasing when the graph is rising from left to right. Let's assume the x - values of the critical points are $a$, $b$, $c$ (from left - to - right). If the graph rises in the intervals $(a,b)$ and $(c,\infty)$ (these are just examples based on general graph analysis), the increasing intervals are the intervals where the slope of the tangent line to the curve is positive.

Step3: Identify decreasing intervals

The function is decreasing when the graph is falling from left to right. For example, if the graph falls in the intervals $(-\infty,a)$ and $(b,c)$ (again, examples based on general graph analysis), the decreasing intervals are the intervals where the slope of the tangent line to the curve is negative.

Step4: Find $x$ - values where $f^{\prime}(x) = 0$

These are the $x$ - coordinates of the local maxima and minima (peaks and valleys) of the graph. At these points, the tangent line to the curve is horizontal, so its slope $f^{\prime}(x)=0$.

Step5: Find critical points

Critical points are of the form $(x,f(x))$ where $x$ is the $x$ - value from the previous step and $f(x)$ is the corresponding $y$ - value on the graph.

Answer:

• Increasing: (Intervals where the graph rises from left - to - right, need to read from the actual graph)
• Decreasing: (Intervals where the graph falls from left - to - right, need to read from the actual graph)
• $x$ values where $f^{\prime}(x)=0$: (X - coordinates of local maxima and minima, need to read from the actual graph)
• Critical points $(x,f(x))$: (Pairs of $x$ - coordinates from the previous step and their corresponding $y$ - values on the graph, need to read from the actual graph)