- Calculate the second derivative: \(S''(t) = -\frac{15}{(t+1)^2} + 3\).
- Find the inflection point by setting \(S''(t) = 0\), which results in \(t = \sqrt{5} - 1 \approx 1.2\).
- Determine concavity by testing the sign of \(S''(t)\): \(S(t)\) is concave down on \([0, 1.2)\) and concave up on \((1.2, \infty)\).
- Inflection point: \(t \approx \boxed{1.2}\).
Explanation
- Calculating the Derivatives
The growth of subscribers is modeled by the function \(S(t) = 15 \ln(t + 1) + \frac{3}{2}t^2\), where \(t \ge 0\). To understand the curvature of this growth, we must examine the concavity of the function, which depends on its second derivative. The first derivative, \(S'(t)\), represents the growth rate, and the second derivative, \(S''(t)\), describes how that rate changes over time. We calculate the derivatives as follows:
First, \(S'(t) = \frac{d}{dt}(15 \ln(t + 1) + \frac{3}{2}t^2) = \frac{15}{t + 1} + 3t\).
Next, \(S''(t) = \frac{d}{dt}(15(t + 1)^{-1} + 3t) = -15(t + 1)^{-2} + 3 = -\frac{15}{(t + 1)^2} + 3\).
- Finding the Inflection Point
To identify intervals of concavity and inflection points, we find where the second derivative is zero.
Setting \(S''(t) = 0\):
\[-\frac{15}{(t + 1)^2} + 3 = 0\]
\[\frac{15}{(t + 1)^2} = 3\]
\[(t + 1)^2 = 5\]
\[t + 1 = \sqrt{5} \quad (\text{since } t \ge 0, \text{ we take the positive square root})\]
\[t = \sqrt{5} - 1 \approx 1.236\]
Rounding to one decimal place, the inflection point occurs at \(t \approx 1.2\).
- Determining Concavity Intervals
Now, we test the sign of \(S''(t)\) in the intervals created by the inflection point \([0, \sqrt{5} - 1)\) and \((\sqrt{5} - 1, \infty)\).
For the interval \([0, \sqrt{5} - 1)\), pick \(t = 0\):
\(S''(0) = -\frac{15}{(0 + 1)^2} + 3 = -15 + 3 = -12 < 0\). Since the second derivative is negative, the function is concave down.
For the interval \((\sqrt{5} - 1, \infty)\), pick \(t = 2\):
\(S''(2) = -\frac{15}{(2 + 1)^2} + 3 = -\frac{15}{9} + 3 = -1.666... + 3 = 1.333... > 0\). Since the second derivative is positive, the function is concave up.
- Final Results
Based on our analysis, the function \(S(t)\) is concave down on the interval \([0, \sqrt{5} - 1)\) and concave up on the interval \((\sqrt{5} - 1, \infty)\). The inflection point occurs at \(t = \sqrt{5} - 1\). Rounding to one decimal place as requested:
AInterval where \(S\) is concave down: \([0, 1.2)\).
BInterval where \(S\) is concave up: \((1.2, \infty)\).
CInflection point: \(t = 1.2\).
Examples
Concavity analysis is essential in business analytics to understand growth trends. For instance, a tech company tracking the number of active users per month can use the concavity of their growth function to identify when user acquisition begins to accelerate (concave up) or slow down relative to their saturation point (concave down). This informs marketing strategies, budget allocation, and infrastructure planning to ensure the platform handles traffic efficiently.