Find the points of inflection and discuss the concavity of the function. $f ( x ) = 7 x ^ { 3 } + 8 x ^ { 2 } + 5 x - 8$
A)inflection point at $x = - \frac { 8 } { 21 }$ ; concave downward on $\left( - \infty , - \frac { 8 } { 21 } \right)$ ; concave upward on $\left( - \frac { 8 } { 21 } , \infty \right)$
B)inflection point at $x = - \frac { 8 } { 21 }$ ; concave upward on $\left( - \infty , - \frac { 8 } { 21 } \right)$ ; concave downward on $\left( - \frac { 8 } { 21 } , \infty \right)$
C)inflection point at $x = \frac { 8 } { 21 }$ ; concave downward on $\left( - \infty , \frac { 8 } { 21 } \right)$ ; concave upward on $\left( \frac { 8 } { 21 } , \infty \right)$
D)inflection point at $x = \frac { 8 } { 21 }$ ; concave upward on $\left( - \infty , \frac { 8 } { 21 } \right)$ ; concave downward on $\left( \frac { 8 } { 21 } , \infty \right)$
E)none of the above

Question

Quizplus · Accepted Answer

To find the points of inflection, we first find the second derivative of $f(x) = 7x^3 + 8x^2 + 5x - 8$, which is $f''(x) = 42x + 16$. Setting $f''(x) = 0$ gives $x = -\frac{8}{21}$. The sign of $f''(x)$ changes from positive to negative at $x = -\frac{8}{21}$, indicating a point of inflection. The function is concave downward for $x < -\frac{8}{21}$ and concave upward for $x > -\frac{8}{21}$.

Find the Points of Inflection and Discuss the Concavity of the Function