For what values of x is $f (x) = 2 x^{3} + 5 x + 12$ $f(x)= 2x^3+5x+12$ concave or convex?

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Answer 1 · 2017-03-07T09:26:35

$f (x)$ $f(x)$ is concave when $x \in] - \infty, 0 [$ $x in ]-oo,0[$
$f (x)$ $f(x)$ is convex when $x \in] 0, + \infty [$ $x in ]0,+oo[$

We calculate the first and second derivatives

$f (x) = 2 x^{3} + 5 x + 12$ $f(x)=2x^3+5x+12$

$f'(x)=6x^2+5$

$f''(x)=12x$

$f'(x)>0$

$f''(x)=0$ , when $x=0$

We draw a chart

$color(white)(aaaa)$ $Interval$ $color(white)(aaaaaaa)$ $]-oo,0[$ $color(white)(aaaa)$ $]0,+oo[$

$color(white)(aaaa)$ $Sign f''(x)$ $color(white)(aaaaaaaa)$ $-$ $color(white)(aaaaaaaa)$ $+$

$color(white)(aaaa)$ $function$ $color(white)(aaaaaaaaaa)$ $nnn$ $color(white)(aaaaaaaa)$ $uuu$

Therefore,

$f(x)$ is concave when $x in ]-oo,0[$

$f(x)$ is convex when $x in ]0,+oo[$

For what values of x is f(x)= 2x^3+5x+12f(x)=2x3+5x+12f(x)= 2x^3+5x+12 concave or convex?