Question

determine any inflection points for the function f(t)=3t^{2}-6e^{t}. enter each inflection point as an ordered pair. separate multiple answers with commas. enter dne if no such points exist. inflection points:

Explanation:

Step1: Find the first - derivative

First, find the derivative of $f(t)=3t^{2}-6e^{t}$. Using the power - rule $(x^{n})^\prime=nx^{n - 1}$ and the rule $(e^{x})^\prime = e^{x}$, we have $f^\prime(t)=6t-6e^{t}$.

Step2: Find the second - derivative

Then, find the derivative of $f^\prime(t)$. Using the sum - difference rule and the same basic rules of differentiation, $f^{\prime\prime}(t)=6 - 6e^{t}$.

Step3: Set the second - derivative equal to zero

To find the inflection points, set $f^{\prime\prime}(t)=0$. So, $6 - 6e^{t}=0$.

Step4: Solve for $t$

First, rewrite the equation as $6e^{t}=6$, then $e^{t}=1$. Since $y = e^{x}$ and $e^{0}=1$, we get $t = 0$.

Step5: Find the $y$ - value of the inflection point

Substitute $t = 0$ into the original function $f(t)$. $f(0)=3(0)^{2}-6e^{0}=0 - 6=-6$.

Answer:

$(0,-6)$