# Is f(x)=sinx concave or convex at x=pi/2?

Oct 24, 2017

See below

#### Explanation:

to even find concavity, you should know what it means by concave up and concave down.

if a functions $f ' ' \left(x\right)$ for a point, like $\left(0.25 , 2.25\right)$ is positive, like this graph of ${x}^{3}$, but in its 2nd derivative form of $9 x$

graph{y=9x [-10, 10, -5, 5]}

then its concave-Up

if a functions $f ' ' \left(x\right)$ for a point, like $\left(0.2 , - 0.54\right)$ is negative, like the graph of $3 {x}^{3} + 2 x + 4$, which will turn into $- 27 x$

graph{-27x [-10, 10, -5, 5]}

then its concave-Down

but all of these points and graphs are linear, your looking for an point on an non-linear graph $\left(\sin \left(x\right)\right)$.

the second derivative of $\sin \left(x\right)$ is $- \sin \left(x\right)$

here's the graph:

graph{-sin(x) [-10, 10, -5, 5]}

at $x = \frac{\pi}{2}$, the point is negative, also known as concave-down. since the graph is going negative at the interval $\left[0 , \pi\right]$