sketch the graph of a function that has the properties described below. (-2, -1), (-4, -4), and (-6, -7) are…
sketch the graph of a function that has the properties described below. (-2, -1), (-4, -4), and (-6, -7) are on the graph. f(-2)=0 and f(-6)=0, f(x)<0 for x > -4, f(-4)=0, f(x)>0 for x < -4. choose the correct graph below.
Answer
Explanation:
Step1: Analyze critical points
The points $(-2,-1)$, $(-4, - 4)$ and $(-6,-7)$ are on the graph. Also, $f'(-2)=0$ and $f'(-6) = 0$ indicate that there are horizontal - tangents at $x=-2$ and $x = - 6$.
Step2: Analyze concavity
Since $f''(x)<0$ for $x>-4$, the function is concave - down for $x > - 4$. Since $f''(x)>0$ for $x<-4$, the function is concave - up for $x < - 4$. And $f''(-4)=0$ is an inflection point.
Step3: Evaluate graphs
Graph A has no horizontal tangents at $x=-2$ and $x=-6$. Graph B is always concave - down. Graph C does not have the correct concavity and horizontal tangents. Graph D has horizontal tangents at appropriate $x$ - values and correct concavity.
Answer:
D.