Concavity From A Second Derivative Table
The table below provides selected values for $$f''(x)$$, the second derivative of the function $$f(x)=x^{3}-6x^{2}+9x+2$$. Based on the information in the table, on which interval is the graph of $$f$$ concave down?
| x | $$f''(x)=6x-12$$ |
|---|---|
| 0 | -12 |
| 2 | 0 |
| 4 | 12 |
A
The function is concave down only at $$x=2$$ where $$f''(x)=0$$.
B
For values of $$x > 2$$, the function is concave down because $$f''(x)$$ is positive.
C
The table does not provide any information about concavity.
D
For values of $$x < 2$$, since $$f''(x)=6*x-12$$ is negative, the function is concave down on $$(-\infty,2)$$.
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