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We know that the sign of the derivative tells us whether a function is increasing or decreasing; for example, when $$f'(x)>0$$, $$f(x)$$ is increasing. The sign of the second derivative $$f''(x)$$ tells us whether $$f'$$ is increasing or decreasing; we have seen that if $$f'$$ is zero and increasing at a point then there is a local minimum at the point, and if $$f'$$ is zero and decreasing at a point then there is a local maximum at the point. Thus, we extracted information about $$f$$ from information about $$f''$$.
We can get information from the sign of $$f''$$ even when $$f'$$ is not zero. Suppose that $$f''(a)>0$$. This means that near $$x=a$$, $$f'$$ is increasing. If $$f'(a)>0$$, this means that $$f$$ slopes up and is getting steeper; if $$f'(a) < 0$$, this means that $$f$$ slopes down and is getting less steep. The two situations are shown in Figure $$\PageIndex{1}$$. A curve that is shaped like this is called concave up.