Is $f (x) = - x^{5} - 2 x^{2} - 6 x + 3$ $f(x)=-x^5-2x^2-6x+3$ concave or convex at $x = - 4$ $x=-4$ ?

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Answer 1 · 2017-09-29T12:05:11

The function is convex

Calculate the first and second derivatives

$f (x) = - x^{5} - 2 x^{2} - 6 x + 3$ $f(x)=-x^5-2x^2-6x+3$

$f'(x)=-5x^4-4x-6$

$f''(x)=-20x^3-4$

Therefore, when $x=-4$

$f''(-4)=-20*(-4)^3-4=1280-4=1276$

As $f''(x)>0$ , the function is convex at $x=-4$

Is f(x)=-x^5-2x^2-6x+3f(x)=−x5−2x2−6x+3f(x)=-x^5-2x^2-6x+3 concave or convex at x=-4x=−4x=-4?