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ST3.xx_Practice_Test_Autumn...
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Question list $\quad K$
Sketch the graph of the given function by determining the appropraate information and points from the first and second derivatives. Use a graphing calculator to check the graph.
\[
y=x^{3}-8 x^{2}-12 x+2
\]
Question 5
What are the coordinates of the relative maxima? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Question 6
A. (Simplify your answer. Type an ordered pair. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
Question 7
B. There is no maximum.
Question 8 What are the coordinates of the relative minima? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. (Simplify your answer. Type an ordered pair. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
B. There is no minimum.
What are the coordinates of the points of inflection? Select the correct choice below and, if necessary. fill in the answer box to complete your choice.
Question 11
Question 12
Question 13
Question 14
Question 15
A. (Simplify your answer. Type an ordered pair. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
B. There are no inflection points.
On what interval(s) is $y$ increasing or decreasing?
A. $y$ is increasing on $x> \frac{8}{3}, y$ is decreasing on $x< \frac{8}{3}$
B. $y$ is increasing on $x< \frac{8}{3}, y$ is decreasing on $x \geq \frac{8}{3}$.
c. $y$ is increasing on $x< -\frac{2}{3}$ and $x> 6$ is decreasing on $-\frac{2}{3}< x< 6$.
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Answer
\(\boxed{\text{Points of inflection: }(\frac{8}{3}, -67.926)}\)
Step 1 ：Find the first and second derivatives of the function: $y' = 3x^2 - 16x - 12$ and $y'' = 6x - 16$.
Step 2 ：Determine the critical points by setting $y'$ equal to zero: $3x^2 - 16x - 12 = 0$. The critical points are $x = -\frac{2}{3}$ and $x = 6$.
Step 3 ：Determine the inflection points by setting $y''$ equal to zero: $6x - 16 = 0$. The inflection point is $x = \frac{8}{3}$.
Step 4 ：Analyze the first and second derivatives to determine the relative maxima, relative minima, and points of inflection.
Step 5 ：Relative maxima: $(-\frac{2}{3}, 6.148)$
Step 6 ：Relative minima: $(6, -142)$
Step 7 ：Points of inflection: $(\frac{8}{3}, -67.926)$
Step 8 ：\(\boxed{\text{Relative maxima: }(-\frac{2}{3}, 6.148)}\)
Step 9 ：\(\boxed{\text{Relative minima: }(6, -142)}\)
Step 10 ：\(\boxed{\text{Points of inflection: }(\frac{8}{3}, -67.926)}\)
