the temperature t (in c) of a cup of tea after being removed from the stove is modeled by t(t)=95e^{-0.6t}+34…

the temperature t (in c) of a cup of tea after being removed from the stove is modeled by t(t)=95e^{-0.6t}+34, 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)=95e^{-0.6t}+34, 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 of $T(t)$

Using the sum rule and the chain - rule, if $T(t)=95e^{- 0.6t}+34$, then $T^\prime(t)=95\times(-0.6)e^{-0.6t}=-57e^{-0.6t}$.

Step2: Find the second - derivative of $T(t)$

Differentiate $T^\prime(t)$ with respect to $t$. Using the chain - rule, $T^{\prime\prime}(t)=-57\times(-0.6)e^{-0.6t}=34.2e^{-0.6t}$.

Step3: Determine concavity and inflection points

Since $T^{\prime\prime}(t)=34.2e^{-0.6t}>0$ for all $t\geq0$ (because the exponential function $y = e^{-0.6t}>0$ for all real $t$ and $34.2>0$), the function $T(t)$ is concave up on the interval $[0,\infty)$. To find the inflection point, we set $T^{\prime\prime}(t) = 0$. But $34.2e^{-0.6t}=0$ has no solution since $e^{-0.6t}>0$ for all $t\in R$.

Answer:

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