Question

Is `f(x)=x^3+2x^2-4x-12` concave or convex at `x=3`?

Answer 1

At `f(3)` the function is convex.

Explanation:

A concave function is a function in which no line segment joining two points on its graph lies above the graph at any point.

A convex function, on the other hand, is a function in which no line segment joining two points on the graph lies below the graph at any point.

It means that, if `f(x)` is more than the average of `f(x+-lambda)` than the function is concave and if `f(x)` is less than the average of `f(x+-lambda)` than the function is convex.

Hence to find the convexity or concavity of `f(x)=x^3+2x^2-4x-12` at `x=3`, let us evaluate `f(x)` at `x=2.5, 3 and 3.5`.

`f(2.5)=(5/2)^3+2(5/2)^2-4(5/2)-12=125/8+50/4-10+12=(125+100-80-96)/8=49/8`

`f(3)=(3)^3+2(3)^2-4(3)-12=27+18-12-12=21`

`f(3.5)=(7/2)^3+2(7/2)^2-4(7/2)-12=343/8+98/4-14-12=(343+196-112-96)/8=331/8`

The average of `f(2.5)` and `f(3.5)` is `(49/8+331/8)/2=380/(2xx8)=95/4=23 3/4`

As, this is more than `f(3)`, at `f(3)` the function is convex.

graph{x^3+2x^2-4x-12 [-5, 5, -20, 30]}