I have the following graph and want to know:
Are these statements correct and if not why?
- question A: $f'(-2) > 0$
- question B: $f(0) = 0$ and $f'(0) = 0$ and $f''(0) \neq 0 $
- question C: there are two intervals where the curvage of the graph is curved right
- question D: $f''(-1) < 0$
- question E: $f(6) > f(-4)$
- question F: $f'(-4) > f'(6)$
- question G: $f'(6) = 0$ and $f''(6) >0$
I am new to this materia (had just some lessons) and need simple explainations.
Currently my solutions are:
- question A: dont know
- question B: correct and correct and dont know
- question C: is correct between $-4$ to $0$ and $0$ to $6$
- question D: dont know
- question E: correct, -2.5 > -3.9
- question F:
- qeustion G:
question A: $f'(-2) > 0$
Yes, it is correct. It looks like there is an instantaneous slope of around $+1$.
question B: $f(0) = 0$ and $f'(0) = 0$ and $f''(0) \neq 0 $
No, it is not correct because the statement $f''(0) \neq 0$ is false. (The statements $f(0)=0$ and $f'(0)=0$ are true, by the way.) From the graph, you will see that concavity changes. The graph is concave down for the interval $(-\infty,0)$, then it is concave up for the interval $(0,2)$.
question C: there are two intervals where the curvage of the graph is curved right
Yes, it is correct. There are two intervals where $f$ is concave down: intervals $(-4,0)$ and $(2,5)$.
question D: $f''(-1) < 0$
Yes, it is correct. The graph is concave down on the interval $(-4,0)$, as I stated above in my response to question B. Since $x=-1$ falls in the interval $(-4,0)$, the statement $f''(x)<0$ is true.
question E: $f(6) > f(-4)$
Yes, it is correct, because as you said, $f(6)=-2.5$ and $f(-4)=-3.9$, and $-2.5 > -3.9$.
question F: $f'(-4) > f'(6)$
No, it is not correct. Like my answer to your other question, $f(-4)=f(6)=0$. So it cannot be true that $f(-4) > f(6)$.
question G: $f'(6) = 0$ and $f''(6) >0$
Yes, it is correct. The graph has a horizontal tangent line at $x=6$; so $f'(6)=0$. The graph is also concave up on the interval $(5,\infty)$. Since $x=6$ falls in this interval, $f''(6)>0$.