Find Concavity and POI

• Aug 5th 2009, 09:50 AM
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....
• Aug 5th 2009, 10:12 AM
skeeter
Quote:

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.