Find the interval(s) where f is concave up for $f ( x ) = 4 x ^ { 4 } - 3 x ^ { 3 } + 5 x - 10$
A) (-?, ?)
B) (-?, 0) ? $\left( \frac { 3 } { 8 } , \infty \right)$
C) $\left( \frac { 3 } { 8 } , \infty \right)$
D) $\left( - \infty , \frac { 3 } { 8 } \right)$
E) $(\left. 0 , \frac { 3 } { 8 } \right)$

Question

Quizplus · Accepted Answer

To determine where the function $f(x) = 4x^4 - 3x^3 + 5x - 10$ is concave up, we need to find the second derivative $f''(x)$ and analyze its sign. The first derivative $f'(x)$ is $16x^3 - 9x^2 + 5$, and the second derivative $f''(x)$ is $48x^2 - 18x$. Factoring out $6x$ gives $f''(x) = 6x(8x - 3)$. Setting $f''(x) = 0$ gives $x = 0$ and $x = \frac{3}{8}$. These are potential inflection points. The sign of $f''(x)$ changes from negative to positive at $x = 0$ and from positive to negative at $x = \frac{3}{8}$, indicating concavity changes. Thus, $f(x)$ is concave up when $x < 0$ and $x > \frac{3}{8}$, which corresponds to the intervals $(-∞, 0)$ and $\left(\frac{3}{8}, ∞\right)$.

Find the Interval(s) Where F Is Concave Up For f(x)=4x4−3x3+5x−10f ( x ) = 4 x ^ { 4 } - 3 x ^ { 3 } + 5 x - 10f(x)=4x4−3x3+5x−10

Find the Interval(s) Where F Is Concave Up For $f ( x ) = 4 x ^ { 4 } - 3 x ^ { 3 } + 5 x - 10$