the temperature t (in c) of a cup of tea after being removed from the stove is modeled by t(t) = 75e^{-0.8t}…

the temperature t (in c) of a cup of tea after being removed from the stove is modeled by t(t) = 75e^{-0.8t} + 11, t ≥ 0. find the intervals of concavity and determine if there is an inflection point. (a) interval(s) where t is concave up (use interval notation): (b) enter the value of the inflection point (if none exists enter \none\): none

the temperature t (in c) of a cup of tea after being removed from the stove is modeled by t(t) = 75e^{-0.8t} + 11, t ≥ 0. find the intervals of concavity and determine if there is an inflection point. (a) interval(s) where t is concave up (use interval notation): (b) enter the value of the inflection point (if none exists enter \none\): none

Answer

Explanation:

Step1: Find the first derivative

Using the chain - rule, if (y = 75e^{-0.8t}+11), then (y^\prime=T^\prime(t)=75\times(- 0.8)e^{-0.8t}=-60e^{-0.8t})

Step2: Find the second derivative

Differentiate (T^\prime(t)) with respect to (t). Using the chain - rule again, if (y=-60e^{-0.8t}), then (y^{\prime\prime}=T^{\prime\prime}(t)=(-60)\times(-0.8)e^{-0.8t}=48e^{-0.8t})

Step3: Analyze the concavity and inflection points

Since (t\geq0) and (e^{-0.8t}=\frac{1}{e^{0.8t}}>0) for all real (t), then (T^{\prime\prime}(t) = 48e^{-0.8t}>0) for all (t\geq0)

Answer:

a) The function (T(t)) is concave up on the interval ([0,\infty)) b) There is no inflection point, so enter "none"