Is $f (x) = - 4 x^{5} - 2 x^{4} - 5 x^{3} + 2 x - 31$ $f(x)=-4x^5-2x^4-5x^3+2x-31$ concave or convex at $x = - 2$ $x=-2$ ?

Question

1361 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-12-03T02:23:37

convex

If $f''(-2)<0$ , the function is concave at that point.
If $f''(-2)>0$ , the function is convex at that point.

$f'(x)=-20x^4-8x^3-15x^2+2$

$f''(x)=-80x^3-24x^2-15x+2$

$f''(-2)=576$

$576>0$ , so $f(x)$ is convex at the point $(-2,101)$ .

graph{-4x^5-2x^4-5x^3+2x-31 [-43, 61, -44.06, 7.94]}

Notice how the graph does appear to be convex (making a $uu$ shape) at $x=-2$ .

Is f(x)=-4x^5-2x^4-5x^3+2x-31f(x)=−4x5−2x4−5x3+2x−31f(x)=-4x^5-2x^4-5x^3+2x-31 concave or convex at x=-2x=−2x=-2?