the temperature t (in c) of a cup of tea after being removed from the stove is modeled by t(t)=120e^(-0.4t)+2…

the temperature t (in c) of a cup of tea after being removed from the stove is modeled by t(t)=120e^(-0.4t)+27, 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\).

the temperature t (in c) of a cup of tea after being removed from the stove is modeled by t(t)=120e^(-0.4t)+27, 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\).

Answer

Explanation:

Step1: Find the first - derivative

We have $T(t)=120e^{- 0.4t}+27$. Using the chain - rule, $T^\prime(t)=120\times(-0.4)e^{-0.4t}=-48e^{-0.4t}$.

Step2: Find the second - derivative

Differentiate $T^\prime(t)$ with respect to $t$. Using the chain - rule again, $T^{\prime\prime}(t)=-48\times(-0.4)e^{-0.4t}=19.2e^{-0.4t}$.

Step3: Analyze concavity and inflection points

Since $T^{\prime\prime}(t)=19.2e^{-0.4t}>0$ for all $t\geq0$ (because the exponential function $e^{-0.4t}>0$ for all real $t$ and $19.2 > 0$), the function $T(t)$ is concave up on the interval $[0,\infty)$. And since $T^{\prime\prime}(t)$ is never equal to $0$ for $t\geq0$, there are no inflection points.

Answer:

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