# Thread: Find Concavity and POI

1. ## Find Concavity and POI

for $\displaystyle y= x^3-9x^2+27x$
from the first derivative $\displaystyle y'=3x^2-18x+27$
If found the critical no x = 3
and attempted the 1st deriv. test (-inf,3)(3,inf)
and tested with points 1,4 and got two positives does this mean I have to try with the 2nd derivative
$\displaystyle y''=6x-18$
I'm confused with what I do next....

for $\displaystyle y= x^3-9x^2+27x$
from the first derivative $\displaystyle y'=3x^2-18x+27$
If found the critical no x = 3
and attempted the 1st deriv. test (-inf,3)(3,inf)
and tested with points 1,4 and got two positives does this mean I have to try with the 2nd derivative
$\displaystyle y''=6x-18$
I'm confused with what I do next....
$\displaystyle y = x^3 - 9x^2 + 27x$

$\displaystyle y' = 3x^2 - 18x + 27$

$\displaystyle y'' = 6x - 18$

critical values for y' ...

$\displaystyle 3x^2 - 18x + 27 = 0$

$\displaystyle x^2 - 6x + 9 = 0$

$\displaystyle (x -3)^2 = 0$

$\displaystyle x = 3$

no extrema at $\displaystyle x = 3$ because y' does not change signs at $\displaystyle x = 3$

critical value for y'' ...

$\displaystyle 6x - 18 = 0$

$\displaystyle x = 3$

y'' changes sign at $\displaystyle x = 3$ ... the point (3, 27) is an inflection point on the original function.

now ... graph the original function and confirm the analysis.