the temperature $t$ (in c) of a cup of tea after being removed from the stove is modeled by…

the temperature $t$ (in c) of a cup of tea after being removed from the stove is modeled by $t(t)=75e^{-0.2t}+34, tgeq0$. 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\):

the temperature $t$ (in c) of a cup of tea after being removed from the stove is modeled by $t(t)=75e^{-0.2t}+34, tgeq0$. 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\):

Answer

Explanation:

Step1: Find the first - derivative

Using the sum rule and the chain - rule, if $T(t)=75e^{-0.2t}+34$, then $T^\prime(t)=75\times(- 0.2)e^{-0.2t}=-15e^{-0.2t}$.

Step2: Find the second - derivative

Differentiate $T^\prime(t)$ with respect to $t$. Using the chain - rule, $T^{\prime\prime}(t)=-15\times(-0.2)e^{-0.2t}=3e^{-0.2t}$.

Step3: Analyze concavity

Since $T^{\prime\prime}(t)=3e^{-0.2t}>0$ for all $t\geq0$ (because the exponential function $y = e^{-0.2t}>0$ for all real $t$ and $3>0$), the function $T(t)$ is concave up on the interval $[0,\infty)$.

Step4: Find the inflection point

An inflection point occurs where $T^{\prime\prime}(t) = 0$ or $T^{\prime\prime}(t)$ is undefined. Since $T^{\prime\prime}(t)=3e^{-0.2t}>0$ for all $t\geq0$, there is no value of $t$ for which $T^{\prime\prime}(t) = 0$. Also, $T^{\prime\prime}(t)$ is defined for all $t\geq0$. So, there is no inflection point.

Answer:

(a) $[0,\infty)$ (b) none