Determine Monotonicity Intervals of Derivative Function

To determine the monotonicity of the derivative function $f'(x)$, we need to examine the sign of its derivative, $f''(x)$, on the interval $(-t, t)$.

Step 1: Relation between $f'(x)$ and $f''(x)$

By definition of monotonicity for a differentiable function:

• $f'(x)$ is increasing on an interval where $f''(x) > 0$.
• $f'(x)$ is decreasing on an interval where $f''(x) < 0$.

Step 2: Sign Analysis of $f''(x)$

Evaluating the second derivative $f''(x)$ over the given domain $(-t, t)$, we find that $x = -\frac{1}{4}$ is a critical point where $f''(x) = 0$:

1. For $x \in \left(-t, -\frac{1}{4}\right)$:
   $f''(x) < 0$
   Since the second derivative $f''(x)$ is negative in this interval, $f'(x)$ is strictly decreasing in $\left(-t, -\frac{1}{4}\right)$.

2. For $x \in \left(-\frac{1}{4}, t\right)$:
   $f''(x) > 0$
   Since the second derivative $f''(x)$ is positive in this interval, $f'(x)$ is strictly increasing in $\left(-\frac{1}{4}, t\right)$.

Conclusion

Thus, $f'(x)$ is decreasing in $\left(-t, -\frac{1}{4}\right)$ and increasing in $\left(-\frac{1}{4}, t\right)$.

This matches Option B.