2. (8 points) given the function shown in the graph below, identify the intervals on which the function…

2. (8 points) given the function shown in the graph below, identify the intervals on which the function appears to be increasing or decreasing.\n• increasing:\n• decreasing:\n• x values where f(x)=0:\n• critical points (x, f(x))=

2. (8 points) given the function shown in the graph below, identify the intervals on which the function appears to be increasing or decreasing.\n• increasing:\n• decreasing:\n• x values where f(x)=0:\n• critical points (x, f(x))=

Answer

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)