a. find the open intervals on which the function is increasing and those on which it is decreasing. b…

a. find the open intervals on which the function is increasing and those on which it is decreasing. b. identify the functions local extreme values, if any, saying where they occur. g(t)= - 4t² - 3t + 3 a. find the open intervals on which the function is increasing. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the function is increasing on the open interval(s) (type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. the function is never increasing. find the open intervals on which the function is decreasing. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the function is decreasing on the open interval(s) (type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. the function is never decreasing. b. find each local maximum, if any. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice (simplify your answers.) a. the function has a local maximum value at one value of t. the maximum value is g( ) =

a. find the open intervals on which the function is increasing and those on which it is decreasing. b. identify the functions local extreme values, if any, saying where they occur. g(t)= - 4t² - 3t + 3 a. find the open intervals on which the function is increasing. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the function is increasing on the open interval(s) (type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. the function is never increasing. find the open intervals on which the function is decreasing. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the function is decreasing on the open interval(s) (type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. the function is never decreasing. b. find each local maximum, if any. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice (simplify your answers.) a. the function has a local maximum value at one value of t. the maximum value is g( ) =

Answer

Explanation:

Step1: Find the derivative

The derivative of $g(t)=-4t^{2}-3t + 3$ using the power - rule $(x^n)'=nx^{n - 1}$ is $g'(t)=-8t-3$.

Step2: Find the critical points

Set $g'(t) = 0$. So, $-8t-3=0$. Solving for $t$ gives $t=-\frac{3}{8}$.

Step3: Test the intervals

Choose a test - point in the interval $(-\infty,-\frac{3}{8})$, say $t=-1$. Then $g'(-1)=-8\times(-1)-3 = 5>0$. So, $g(t)$ is increasing on $(-\infty,-\frac{3}{8})$. Choose a test - point in the interval $(-\frac{3}{8},\infty)$, say $t = 0$. Then $g'(0)=-8\times0-3=-3<0$. So, $g(t)$ is decreasing on $(-\frac{3}{8},\infty)$.

Step4: Find local extreme values

Since $g(t)$ changes from increasing to decreasing at $t = -\frac{3}{8}$, the function has a local maximum at $t=-\frac{3}{8}$. Substitute $t = -\frac{3}{8}$ into $g(t)$: $g(-\frac{3}{8})=-4\times(-\frac{3}{8})^{2}-3\times(-\frac{3}{8})+3=-4\times\frac{9}{64}+\frac{9}{8}+3=-\frac{9}{16}+\frac{18}{16}+3=\frac{-9 + 18}{16}+3=\frac{9}{16}+3=\frac{9+48}{16}=\frac{57}{16}$.

Answer:

a. The function is increasing on the open interval $(-\infty,-\frac{3}{8})$. The function is decreasing on the open interval $(-\frac{3}{8},\infty)$. b. The function has a local maximum value at one value of $t$. The maximum value is $g(-\frac{3}{8})=\frac{57}{16}$.