- Determine the first derivative \(T'(t) = -60 e^{-0.6t}\) and the second derivative \(T''(t) = 36 e^{-0.6t}\).
- Observe that \(T''(t) > 0\) for all \(t \geq 0\), which indicates that the function is concave up on the entire domain.
- Conclude that since \(T''(t)\) never equals zero, there is no inflection point.
- Final answer: Concave up on \([0, \infty)\); Inflection point is \(\boxed{\text{none}}\).
Explanation
- Problem Analysis and Definitions
The temperature of the tea is modeled by the function \(T(t) = 100 e^{-0.6t} + 31\), defined for \(t \geq 0\). Concavity of a function is determined by the sign of its second derivative, \(T''(t)\). If \(T''(t) > 0\), the function is concave up; if \(T''(t) < 0\), it is concave down. An inflection point occurs where the concavity changes, which requires \(T''(t) = 0\) or for \(T''(t)\) to be undefined, with a sign change at that point.
- Calculating Derivatives
First, we calculate the first derivative of the temperature function with respect to time \(t\):
\[T'(t) = \frac{d}{dt}(100 e^{-0.6t} + 31) = 100 \times (-0.6) e^{-0.6t} = -60 e^{-0.6t}\]
Next, we find the second derivative by differentiating the first derivative:
\[T''(t) = \frac{d}{dt}(-60 e^{-0.6t}) = -60 \times (-0.6) e^{-0.6t} = 36 e^{-0.6t}\]
- Determining Concavity and Inflection Points
We analyze the sign of the second derivative \(T''(t) = 36 e^{-0.6t}\) for \(t \geq 0\). Since the exponential function \(e^x\) is always positive for any real number \(x\), and \(36\) is positive, it follows that \(36 e^{-0.6t} > 0\) for all \(t \geq 0\). Because \(T''(t)\) is strictly positive on its entire domain, the function \(T(t)\) is concave up on the interval \([0, \infty)\). Since \(T''(t)\) is never zero and never changes sign, there are no points of inflection.
- Final Answer
Based on the analysis, the function is concave up on the interval \([0, \infty)\) and has no inflection points.
AInterval(s) where \(T\) is concave up: \(\boxed{[0, \infty)}\)
BInflection point value: \(\boxed{\text{none}}\)
Examples
In thermal physics and engineering, monitoring the rate at which an object cools is essential for safety and quality control. For instance, if you are cooling down a machine part after heat treatment, analyzing the concavity of the temperature curve helps determine how the rate of cooling changes over time, ensuring the material properties are consistent as it reaches ambient temperature. Applying derivatives to understand these curves allows engineers to predict cooling behavior accurately.