Is f(x)=10x^5-7x^4+x-4 concave or convex at x=-1?

Jul 20, 2017

f is concave at $x = - 1$

Explanation:

We will find the second derivative and substitute the given value of x; if $f ' ' \left(- 1\right) > 0$ then it is convex:

$f ' \left(x\right) = 50 {x}^{4} - 28 {x}^{3} + 1$

$f ' ' \left(x\right) = 200 {x}^{3} - 84 {x}^{2}$

$f ' ' \left(- 1\right) = - 200 - 84 < 0$

Then f is concave at $x = - 1$
graph{10x^5-7x^4+x-4 [-5, 2, -40, 2]}