Question

Is `f(x)=5x^3-2x^2+5x+12` concave or convex at `x=-1`?

Answer 1

The function is concave at `f(-1)`

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)=5x^3-2x^2+5x+12` at `x=-1`, let us evaluate `f(x)` at `x=-1.5, -1 and -0.5`.

`f(-1.5)=5(-3/2)^3-2(-3/2)^2+5(-3/2)+12=-135/8-45/4-15/2+12=(-135-90-60+96)/8=-189/8`

`f(-1)=5(-1)^3-2(-1)^2+5(-1)+12=-5-2-5+12=0`

`f(-0.5)=5(-1/2)^3-2(-1/2)^2+5(-1/2)+12=-5/8-2/4-5/2+12=(-5-4-20+96)/8=67/8`

The average of `f(-1.5)` and `f(-0.5)` is `(-189/8+67/8)/2=-122/(2xx8)=-61/8`

As, this is less than `f(-1)`, at `f(-1)` the function is concave.

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