Question28 Mark for Review
The first derivative of the function $h$ is given by $h^{\prime}(x)=x^{4}-x^{3}+x$. On which of the following intervals is the graph of $h$ concave down? (A) $(-0.755,0)$ (B) $(0,0.5)$ only (C) $(-0.455, \infty)$ (D) $(-\infty,-0.455)$

Studdy Solution
Analyze the intervals determined by the roots to see where $4x^{3} - 3x^{2} + 1 < 0$.
Test intervals:
1. $(-\infty, -0.455)$
2. $(-0.455, 0.755)$
3. $(0.755, \infty)$
By testing values in each interval, we find that:
- $4x^{3} - 3x^{2} + 1 < 0$ for $x \in (-\infty, -0.455)$
The graph of $h$ is concave down on the interval $(-\infty, -0.455)$.
The correct answer is $\boxed{D}$.