Question

Is `f(x)=x^3-x^2+x-4` concave or convex at `x=-1`?

Answer 1

`"concave at " x=-1`

Explanation:

To determine if a function is concave/convex at f ( a), we require to find the value of f'' ( a)

`• " If " f''(a)>0" then "f(x)" is convex at x=a"`

`• " If " f''(a)<0" then " f(x)" is concave at x=a"`

`f(x)=x^3-x^2+x-4`

`rArrf'(x)=3x^2-2x+1`

`rArrf''(x)=6x-2`

`"and "f''(-1)=-6-2=-8<0`

`rArrf(x)" is concave at "x=-1`
graph{x^3-x^2+x-4 [-10, 10, -5, 5]}