# Find Concavity and POI

• August 5th 2009, 09:50 AM
Find Concavity and POI
for $y= x^3-9x^2+27x$
from the first derivative $y'=3x^2-18x+27$
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
$y''=6x-18$
I'm confused with what I do next....
• August 5th 2009, 10:12 AM
skeeter
Quote:

for $y= x^3-9x^2+27x$
from the first derivative $y'=3x^2-18x+27$
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
$y''=6x-18$
I'm confused with what I do next....

$y = x^3 - 9x^2 + 27x$

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

$y'' = 6x - 18$

critical values for y' ...

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

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

$(x -3)^2 = 0$

$x = 3$

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

critical value for y'' ...

$6x - 18 = 0$

$x = 3$

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

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