Chapter 5 - Graphs and the Derivative - 5.3 Higher Derivatives, Concavity, and the Second Derivative Test - 5.3 Exercises - Page 284: 34

$$ f(x)=8-6x-x^{2} $$ The graph $f(x)$ is always concave downward for all $x$. The graph $f(x)$ has no inflection points.

$$ f(x)=8-6x-x^{2} $$ The first derivative is $$ f^{\prime}(x)=-6-2x, $$ and the second derivative is $$ f^{\prime\prime}(x)=-2, $$ We see that $f^{\prime\prime}(x) \lt 0$ for all $x$, so, $f(x)$ is concave downward for all $x$. Since $f(x)$ is always concave downward , however, so it has no inflection points