consider the function shown in the following graph. the function is defined for all real numbers. note: you…
consider the function shown in the following graph. the function is defined for all real numbers. note: you can get an enlarged view of the graph by clicking on it. use interval notation for both parts below. for what x values is the function increasing? for what x values is the function decreasing? note: you can earn partial credit on this problem.
Answer
Explanation:
Step1: Recall increasing - decreasing concept
A function $y = f(x)$ is increasing when the slope of the tangent line is positive and decreasing when the slope of the tangent line is negative. On a graph, we can visually identify intervals where the function is going up (increasing) or going down (decreasing).
Step2: Identify increasing intervals
Looking at the graph, we see the function is increasing when it is going up from left - to - right. If we assume the critical points (where the function changes from increasing to decreasing or vice - versa) are at $x=a$, $x = b$, $x=c$ etc. Let's say the function is increasing on the intervals $(-\infty,a)$ and $(b,c)$.
Step3: Identify decreasing intervals
The function is decreasing when it is going down from left - to - right. For example, on the intervals $(a,b)$ and $(c,\infty)$.
Since we don't have the actual graph values, assume the critical points are $x_1,x_2,x_3$ such that the function is increasing on the intervals $(-\infty,x_1)\cup(x_2,x_3)$ and decreasing on the intervals $(x_1,x_2)\cup(x_3,\infty)$.
Answer:
Increasing: $(-\infty,x_1)\cup(x_2,x_3)$; Decreasing: $(x_1,x_2)\cup(x_3,\infty)$ (Note: You need to substitute the actual $x$ - values from the graph for $x_1,x_2,x_3$)