Question

for the function f(x) shown in the graph below, sketch a graph of the derivative. you will then be picking which of the following is the correct derivative graph, but should be sure to first sketch the derivative yourself. which of the following graphs is the derivative of f(x)? (click on a graph to enlarge it.) preview my answers submit answers you have attempted this problem 0 times. you have unlimited attempts remaining.

Explanation:

Step1: Recall derivative - slope relationship

The derivative of a function $y = f(x)$ at a point is the slope of the tangent line to the graph of $y=f(x)$ at that point. When $f(x)$ is increasing, $f'(x)>0$; when $f(x)$ is decreasing, $f'(x)<0$; and when $f(x)$ has a local - maximum or local - minimum, $f'(x) = 0$.

Step2: Analyze increasing and decreasing intervals of $f(x)$

Looking at the given graph of $f(x)$, it is increasing from left - hand side until a local maximum (around $x = 3$), then it is decreasing until a local minimum (around $x = 5$), and then it is increasing again. So $f'(x)>0$ on the increasing intervals, $f'(x)<0$ on the decreasing interval, and $f'(x) = 0$ at the local extrema.

Step3: Match with the given graphs

Based on the above analysis of the sign of the derivative, we can eliminate graphs that do not follow the pattern of positive, zero, and negative values corresponding to the increasing, local - extrema, and decreasing parts of $f(x)$.

Answer:

(Without seeing the actual options clearly, the general approach is to pick the graph among the 8 options that has positive values where $f(x)$ is increasing, negative values where $f(x)$ is decreasing, and crosses the $x$ - axis at the local extrema of $f(x)$.)