Determine the point of inflection and discuss the concavity of the graph of ,f(x)=x^4-4x^3?

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Answer 1 · 2018-07-15T12:01:09

$x=0,y=0$ and $x=2,y=-14$

$f'(x)=4x^3-12x^2$

$f''(x)=12x^2-24x$

$f'''(x)=24x-24$
solving $f''(x)=0$ we get
$x(12x-24)=0$

so $x=0$ or $x=2$
and

$f'''(0)=-24$ there is a change from convex to concave

$f'''(2)=24$ there is a change from concave to convex.
further we have

$f(2)=-14$

$f(0)=0$