Question

determine the intervals on which the graph of the function (g(t)=2t^{4}+12t^{3}-9t - 2) is concave up and concave down. separate multiple intervals with commas. concave up: concave down:

Explanation:

Step1: Find the first - derivative

Using the power rule $(x^n)'=nx^{n - 1}$, we have $g'(t)=8t^{3}+36t^{2}-9$.

Step2: Find the second - derivative

Differentiate $g'(t)$ again. $g''(t)=24t^{2}+72t$.

Step3: Set $g''(t) = 0$

$24t^{2}+72t=24t(t + 3)=0$. Solving $24t(t + 3)=0$, we get $t = 0$ and $t=-3$.

Step4: Test intervals

Choose test points in the intervals $(-\infty,-3)$, $(-3,0)$ and $(0,\infty)$.
For the interval $(-\infty,-3)$, let $t=-4$. Then $g''(-4)=24\times(-4)^{2}+72\times(-4)=24\times16-288 = 384 - 288=96>0$.
For the interval $(-3,0)$, let $t=-1$. Then $g''(-1)=24\times(-1)^{2}+72\times(-1)=24 - 72=-48<0$.
For the interval $(0,\infty)$, let $t = 1$. Then $g''(1)=24\times1^{2}+72\times1=24 + 72 = 96>0$.

Answer:

Concave up: $(-\infty,-3),(0,\infty)$
Concave down: $(-3,0)$