GROVE HSC 2U EX 2.6 Q5 POINT OF INFLEXION

QUESTION: Show that the curve $f(x)=2x^3-5$ has an inflexion and find its coordinates.

$f'(x)=6x^2$
$f''(x)=12x$

For possible inflexions: solve $f''(x)=0$.

$12x=0$
$\therefore x=0$.
When $x=0, f(0)=2(0)^3-5=-5$.

So, possible inflexion is $(0,-5)$.

Check concavity to verify inflexion.

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PS. the graph wasn't asked for but here it is anyway :-)