# Second derivative

> 二阶导数描述凹凸性和拐点

Concavity 凹凸性

inflection 拐点

This is a part graph about $$f(x)=e^x$$

![](https://2889346953-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Le4Kw7maZEabKFjsC7z%2F-Lf5hWWAPXrS6pkg310z%2F-Lf5hopc5mu-HGGXwX2c%2Flog_concavity.png?generation=1558112064786458\&alt=media)

Every red line's slop represented as a certain first derivative $$f^\prime(x)$$.

You can see it clearly that it indeed goes steeper and steeper, $$f^\prime(x)$$ gets bigger and bigger.

That means $$f^\prime(x)$$ keep increasing in this interval.

So according to the first derivative theory, $$f^{\prime\prime}(x)$$ must be positive.

For a function interval has a feature of $$f^{\prime\prime}(x) > 0$$, we call it `concave upwards` (凸).

This is a part graph about $$f(x)=-x^2$$

![](https://2889346953-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Le4Kw7maZEabKFjsC7z%2F-Lf5hWWAPXrS6pkg310z%2F-Lf5hopjHhwnGbqC_vV1%2F-x%5E2_concavity.png?generation=1558112075898761\&alt=media)

Every red line was becoming more and more gentle, $$f^\prime(x)$$ gets smaller and smaller.

That means $$f^\prime(x)$$ keep decreasing in this interval.

So according to the first derivative theory, $$f^{\prime\prime}(x)$$ must be negative.

For a function interval has a feature of $$f^{\prime\prime}(x) < 0$$, we call it `concave downwards` (凹).

$$f^\prime(x)$$ first decreasing , then increasing, you'll get an inflection

$$f^\prime(x)$$ first increasing, then decreasing, you'll also get an inflection

![](https://2889346953-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Le4Kw7maZEabKFjsC7z%2F-Lf5hWWAPXrS6pkg310z%2F-Lf5hopsrjgEB_1Z7slM%2Fsin_1to2_derivative.png?generation=1558112065160389\&alt=media)