$\text { Determine the open intervals on which the graph of } y = - 6 x ^ { 3 } + 8 x ^ { 2 } + 6 x - 5 \text { is }$ concave downward or concave upward.
A) concave downward on $( - \infty , \infty )$
B) concave upward on $\left( - \infty , - \frac { 4 } { 9 } \right) ;$ concave downward on $\left( - \frac { 4 } { 9 } , \infty \right)$
C) concave upward on $\left( - \infty , \frac { 4 } { 9 } \right) ;$ concave downward on $\left( \frac { 4 } { 9 } , \infty \right)$
D) concave downward on $\left( - \infty , - \frac { 4 } { 9 } \right) ;$ concave upward on $\left( - \frac { 4 } { 9 } , \infty \right)$
E) concave downward on $\left( - \infty , \frac { 4 } { 9 } \right) ;$ concave upward on $\left( \frac { 4 } { 9 } , \infty \right)$

Question

$\text { Determine the open intervals on which the graph of } y = - 6 x ^ { 3 } + 8 x ^ { 2 } + 6 x - 5 \text { is }$ concave downward or concave upward. 
A) concave downward on $( - \infty , \infty )$
B) concave upward on $\left( - \infty , - \frac { 4 } { 9 } \right) ;$ concave downward on $\left( - \frac { 4 } { 9 } , \infty \right)$
C) concave upward on $\left( - \infty , \frac { 4 } { 9 } \right) ;$ concave downward on $\left( \frac { 4 } { 9 } , \infty \right)$
D) concave downward on $\left( - \infty , - \frac { 4 } { 9 } \right) ;$ concave upward on $\left( - \frac { 4 } { 9 } , \infty \right)$
E) concave downward on $\left( - \infty , \frac { 4 } { 9 } \right) ;$ concave upward on $\left( \frac { 4 } { 9 } , \infty \right)$

Quizplus · Accepted Answer

To determine concavity, we find the second derivative of $y = -6x^3 + 8x^2 + 6x - 5$. The first derivative is $y' = -18x^2 + 16x + 6$, and the second derivative is $y'' = -36x + 16$. Setting $y'' = 0$ gives $x = \frac{4}{9}$. The sign of $y''$ changes from positive to negative as $x$ increases through $\frac{4}{9}$, indicating the graph is concave upward on $\left(-\infty, \frac{4}{9}\right)$ and concave downward on $\left(\frac{4}{9}, \infty\right)$.

Determine the open intervals on which the graph of y=−6x3+8x2+6x−5 is \text { Determine the open intervals on which the graph of } y = - 6 x ^ { 3 } + 8 x ^ { 2 } + 6 x - 5 \text { is } Determine the open intervals on which the graph of y=−6x3+8x2+6x−5 is

$\text { Determine the open intervals on which the graph of } y = - 6 x ^ { 3 } + 8 x ^ { 2 } + 6 x - 5 \text { is }$