Question

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

Answer 1

See below

Explanation:

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

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

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

then its concave-Up

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

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 `(sin(x))`.

the second derivative of `sin(x)` is `-sin(x)`

here's the graph:

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

at `x=pi/2`, the point is negative, also known as concave-down. since the graph is going negative at the interval `[0, pi]`