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 function behavior rules

A function $y = f(x)$ is increasing when $f'(x)>0$ (graph rises from left - to - right) and decreasing when $f'(x)<0$ (graph falls from left - to - right). Critical points occur where $f'(x) = 0$ or $f'(x)$ is undefined.

Step2: Identify increasing intervals

By observing the graph, the function is increasing when the graph is rising. Let's assume the $x$ - axis has values from left to right. If we look at the graph, the function is increasing on 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. That is, when the slope of the tangent line to the curve is negative.

Step4: Find $x$ - values where $f'(x)=0$

These are the points where the slope of the tangent line to the curve is zero, i.e., the horizontal tangent points.

Step5: Determine critical points

Critical points are of the form $(x,f(x))$ where $x$ is the $x$ - value from the $x$ - values where $f'(x) = 0$ or $f'(x)$ is undefined.

Since the graph is not provided with numerical values on the $x$ - axis:

• Increasing: Intervals where the graph rises from left - to - right.
• Decreasing: Intervals where the graph falls from left - to - right.
• $x$ values where $f'(x)=0$: $x$ - coordinates of the points with horizontal tangents.
• Critical points $(x,f(x))$: Points where $f'(x) = 0$ or $f'(x)$ is undefined, in the form $(x,f(x))$ where $x$ is the relevant $x$ - value and $f(x)$ is the corresponding $y$ - value.

If we assume the graph has local minima and maxima at $x = a,b,c$ (from left - to - right) where the horizontal tangents occur:

Answer:

• Increasing: Intervals between local minima and local maxima (e.g., $(a,b)$ and $(c,+\infty)$ if the right - most part of the graph is rising).
• Decreasing: Intervals between local maxima and local minima (e.g., $(-\infty,a)$ and $(b,c)$ if the left - most part of the graph is falling).
• $x$ values where $f'(x)=0$: $x=a,x = b,x = c$ (assuming $a,b,c$ are the $x$ - coordinates of the local extrema).
• Critical points $(x,f(x))$: $(a,f(a)),(b,f(b)),(c,f(c))$