The question is:
The following diagram shows part of the graph of $y = f(x)$. The graph has a local maximum at $A$, where $x = -2$, and a local minimum at $B$, where $x = 6$.
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a) sketch the graph of $y = f’(x)$ on the grid below.
I’m confused how to find and sketch the derivative just knowing the first function’s graph. I know on the first derivative, where x=0 there’s a minimum and maximum. So, will A and B be on the x axis on the grid? How do we figure he rest out?
Write down the following in order from least to greatest: $f(0), f’(6), f’’(-2)$.
How do we figure this out when we don’t know the function?
Best Answer
I will only give general pointers for (a), because this is something that can't be easily expressed in words. $x=-2$ and $x=6$ do correspond to zeros on your graph for $f'(x)$. As $x=-2$ is a local maximum, $f'(x)$ should be decreasing going through $x=-2$; conversely, as $x=6$ is a local minimum, $f'(x)$ is increasing there. Overall you should get for $f'(x)$ a U-shaped curve meeting the $x$-axis at $x=-2$ and $x=6$, with its minimum at roughly $x=2$, corresponding to the inflection point of $f(x)$.
For (b), $f'(6)=0$ because $x=6$ is given as a stationary point (local minimum) of $f(x)$. $f(0)>0$, as the graph of $f(x)$ verifies, while $f''(-2)<0$ because $x=-2$ is a local maximum of $f(x)$ (and thus $f'(x)$ is decreasing at $x=-2$). Thus we have $f''(-2)<f'(6)<f(0)$ by comparing signs, without knowing precisely what $f''(-2)$ or $f(0)$ are.