For what values of x is $f(x)=4x^5-5x^4$ concave or convex?

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Answer 1 · 2017-01-29T10:31:07

The answer is $f(x)$ is concave down for $x in ]-oo, 1]$ and concave up when $x in [1, +oo[$

We calculate the first derivative and we build a sign chart

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

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

$f'(x)=20x^3(x-1)$

The critical points are when $f'(x)=0$

$20x^3(x-1)=0$

$x=0$ and $x=1$

Now we construct the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaaaaa)$ $0$ $color(white)(aaaaaa)$ $1$ $color(white)(aaaaa)$ $+oo$

$color(white)(aaaa)$ $x$ $color(white)(aaaaaaaa)$ $-$ $color(white)(aa)$ #color(white)(aaaa)+# $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $x-1$ $color(white)(aaaaa)$ $-$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $f'(x)$ $color(white)(aaaaa)$ $+$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaa)$ $↗^(0)$ $color(white)(aaaa)$ $↘_(-1)$ $color(white)(aaaa)$ $↗^(+oo)$

Therefore,

$f(x)$ is concave down for $x in ]-oo, 1]$ and concave up when

$x in [1, +oo[$

For what values of x is f(x)=4x^5-5x^4f(x)=4x^5-5x^4 concave or convex?