How do you determine whether the function $f(x)=x^(4)-6x^(3)$ is concave up or concave down and its intervals?

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Answer 1 · 2015-09-07T01:52:25

See the explanation.

$f(x)=x^(4)-6x^(3)$

$f'(x) = 4x^3-18x^2$

$f''(x) = 12x^2-36x = 12x(x-3)$

$f''(x) = 0$ at $0$ and at $3$

On $(-oo,0)$ , we have $f''(x) >0$ so the graph of $f$ is concave up.

On $(0,3)$ , we have $f''(x) <0$ so the graph of $f$ is concave down.

On $(3,-oo)$ , we have $f''(x) >0$ so the graph of $f$ is concave up.

Inflection points are $(0,0)$ and $(3, -81)$

How do you determine whether the function f(x)=x^(4)-6x^(3)f(x)=x^(4)-6x^(3) is concave up or concave down and its intervals?