Since the denominator of $ C'(t) $ is always positive, the sign of $ C'(t) $ depends on the numerator. The expression $ -3t^2 - 4t + 12 $ is positive between the roots. Therefore, $ C(t) $ is increasing for:

["Understanding the Sign of $ C'(t) $: How the Numerator Determines Growth of $ C(t) $", "When analyzing the behavior of a function defined by $ C(t) $, particularly its derivative $ C'(t) $, understanding the sign of the derivative is crucial. One key insight is that since the denominator of $ C'(t) $ is always positive, the sign of $ C'(t) $ depends entirely on the numerator. This principle holds true in many calculus applications, especially when dealing with quadratic expressions.", "Consider the quadratic expression commonly encountered in derivatives like $ -3t^2 - 4t + 12 $. The numerator determines whether $ C'(t) $ is positive, negative, or zero—directly shaping the monotonicity of $ C(t) $.", "### The Role of the Denominator", "In differential expressions, functions like $ C'(t) $ typically take the form:", "$$\nC'(t) = \frac{N(t)}{D(t)}\n$$", "where $ N(t) $ is the numerator and $ D(t) $ is the denominator. When $ D(t) > 0 $ over an interval, the sign of $ C'(t) $ matches the sign of $ N(t) $ in that region.", "### Analyzing the Numerator: $ -3t^2 - 4t + 12 $", "We focus first on the quadratic numerator:", "$$\nN(t) = -3t^2 - 4t + 12\n$$", "This is a downward-opening parabola (since the coefficient of $ t^2 $ is negative). Its roots divide the real line into intervals where the sign of $ N(t) $ changes.", "To find the roots, solve:", "$$\n-3t^2 - 4t + 12 = 0\n$$", "Multiply through by $-1$ to simplify:", "$$\n3t^2 + 4t - 12 = 0\n$$", "Apply the quadratic formula:", "$$\nt = \frac{-4 \pm \sqrt{4^2 - 4(3)(-12)}}{2(3)} = \frac{-4 \pm \sqrt{16 + 144}}{6} = \frac{-4 \pm \sqrt{160}}{6}\n$$", "Simplify $ \sqrt{160} = \sqrt{16 \cdot 10} = 4\sqrt{10} $:", "$$\nt = \frac{-4 \pm 4\sqrt{10}}{6} = \frac{-2 \pm 2\sqrt{10}}{3}\n$$", "So the two roots are:", "$$\nt_1 = \frac{-2 - 2\sqrt{10}}{3}, \quad t_2 = \frac{-2 + 2\sqrt{10}}{3}\n$$", "Approximating $ \sqrt{10} \approx 3.16 $, we get:", "$$\nt_1 \approx \frac{-2 - 6.32}{3} = \frac{-8.32}{3} \approx -2.77\n$$\n$$\nt_2 \approx \frac{-2 + 6.32}{3} = \frac{4.32}{3} \approx 1.44\n$$", "Thus, $ N(t) > 0 $ between the roots $ \left( \frac{-2 - 2\sqrt{10}}{3}, \frac{-2 + 2\sqrt{10}}{3} \right) $, and $ N(t) < 0 $ outside this interval.", "### Implications for $ C'(t) $", "Because the denominator $ D(t) $ is always positive, we conclude:", "- $ C'(t) > 0 $ when $ N(t) > 0 $ → on the interval between the roots: $ t \in \left( \frac{-2 - 2\sqrt{10}}{3}, \frac{-2 + 2\sqrt{10}}{3} \right) $\n- $ C'(t) < 0 $ when $ N(t) < 0 $ → outside the roots", "Therefore, $ C(t) $ is increasing wherever $ C'(t) > 0 $, which occurs between the two roots.", "### Conclusion", "The sign of $ C'(t) $ depends entirely on the numerator $ -3t^2 - 4t + 12 $, since its denominator is always positive. As the numerator is positive between its two real roots, $ C(t) $ increases exactly on that interval.", "So, $ C(t) $ is increasing for:", "$$\nt \in \left( \frac{-2 - 2\sqrt{10}}{3}, \frac{-2 + 2\sqrt{10}}{3} \right)\n$$", "This demonstrates the powerful principle that in rational functions with positive denominators, the derivative’s sign—and hence the function’s monotonicity—is governed solely by the numerator’s sign.", "---", "Key Takeaways:", "- The denominator’s positivity ensures consistent behavior for sign analysis.\n- The numerator’s sign determines $ C'(t) $’s sign.\n- Quadratic expressions with negative leading coefficients produce sign changes between roots.\n- Understanding these patterns simplifies solving growth and decay in calculus.", "If you’re studying functions defined by fractional derivatives, always factor the numerator and analyze its sign—this method reveals critical information about the function’s behavior without full integration."]