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

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Answer 1 · 2018-03-22T07:28:54

The function is concave.

Calculate the first and second derivatives

$f (x) = 5 x^{5} - x^{2} + 4 x + 3$ $f(x)=5x^5-x^2+4x+3$

$f'(x)=25x^4-2x+4$

$f''(x)=100x^3-2$

When $x=-4$ ,

$f''(-4)=100*(-4)^3-2$

$=-6402$

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

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