Question

trace or copy the graph of the given function f. sketch the graph of f on the same coordinate axes.

Explanation:

Step1: Recall derivative - slope relationship

The derivative $f'(x)$ at a point is the slope of the tangent line to the graph of $y = f(x)$ at that point.

Step2: Analyze increasing and decreasing intervals

When $f(x)$ is increasing, $f'(x)>0$ (graph of $f'$ is above the $x$ - axis). When $f(x)$ is decreasing, $f'(x)<0$ (graph of $f'$ is below the $x$ - axis).

Step3: Locate critical points

Critical points of $f(x)$ occur where $f'(x) = 0$ (graph of $f'$ crosses the $x$ - axis). At local maxima and minima of $f(x)$, the slope of the tangent is 0, so $f'(x)=0$.

Step4: Sketch the derivative graph

Based on the above - mentioned rules, start from the left - hand side of the graph of $f(x)$. As $f(x)$ is initially increasing, $f'(x)>0$. When $f(x)$ reaches a local maximum, $f'(x)$ crosses the $x$ - axis from positive to negative. Then as $f(x)$ is decreasing, $f'(x)<0$, and so on.

(Note: Since this is a sketching problem and no specific function is given in equation form, the above steps are general guidelines for sketching the derivative graph. The actual sketch should be made by following the increasing - decreasing nature and critical points of the given graph of $f(x)$.)

Answer:

The correct way to sketch $f'$ is to make sure that the sign of $f'$ (above or below the $x$ - axis) corresponds to the increasing or decreasing nature of $f(x)$ and that $f'$ crosses the $x$ - axis at the critical points of $f(x)$. Without seeing the exact options in a multiple - choice format, the key is to have a graph of $f'$ such that it is positive when $f(x)$ is increasing and negative when $f(x)$ is decreasing and has $x$ - intercepts at the local maxima and minima of $f(x)$.