Is $f (x) = - 2 x^{5} + 7 x^{2} - 6 x + 3$ $f(x)=-2x^5+7x^2-6x+3$ concave or convex at $x = - 4$ $x=-4$ ?

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Answer 1 · 2016-02-25T16:12:00

Convex.

To test a function's concavity or convexity, find the value of the second derivative of the function at a point:

To find the function's second derivative, use the power rule twice:

$f(x)=-2x^5+7x^2-6x+3$

$f'(x)=-10x^4+14x-6$

$f''(x)=-40x^3+14$

The value of the second derivative at $x=-4$ is

$f''(-4)=-40(-4)^3+14=-40(-64)+14=2574$

Since this is $>0$ , the function is convex at $x=-4$ .

Is f(x)=-2x^5+7x^2-6x+3f(x)=−2x5+7x2−6x+3f(x)=-2x^5+7x^2-6x+3 concave or convex at x=-4x=−4x=-4?