The graph of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x - 5$ is concave up for $x

Question

The graph of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x - 5$ is concave up for $x < \frac { 2 } { 3 }$ .

Quizplus · Accepted Answer

The concavity of a graph is determined by the sign of the second derivative. For $f(x) = 3x^4 - 2x^3 - 12x^2 + 18x - 5$, the second derivative $f''(x) = 36x^2 - 12x - 24$. The sign of $f''(x)$ determines the concavity, and without solving for the roots of $f''(x) = 0$, we cannot directly conclude the concavity for $x < \frac{2}{3}$ without further analysis.

The Graph Of f(x)=3x4−2x3−12x2+18x−5f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x - 5f(x)=3x4−2x3−12x2+18x−5

The Graph Of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x - 5$