Question

find the intervals on which the graph of ( f ) is concave upward, the intervals on which the graph of ( f ) is concave downward, and the inflection points.
( f(x)=-x^{6}+12 x^{5}-12 x + 3 )
for what interval(s) of ( x ) is the graph of ( f ) concave upward? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. ( (0,8) )
(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)
b. the graph is never concave upward.
for what interval(s) of ( x ) is the graph of ( f ) concave downward? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. ( (-infty, 0),(8, infty) )
(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)
b. the graph is never concave downward.
determine the ( x ) coordinates of any inflection points of the graph of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your chosen
a. ( x= ) (type an exact answer. use a comma to separate answers as needed)
b. there are no inflection points.

Explanation:

Step1: Find the second derivative

First, find the first derivative of \(f(x)=-x^{6}+12x^{5}-12x + 3\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).
\(f^\prime(x)=-6x^{5}+60x^{4}-12\).
Then find the second derivative:
\(f^{\prime\prime}(x)=-30x^{4}+240x^{3}=-30x^{3}(x - 8)\).

Step2: Determine concavity

Set \(f^{\prime\prime}(x)=0\), so \(-30x^{3}(x - 8)=0\), which gives \(x = 0\) and \(x = 8\).
Test intervals:

• For \(x\lt0\), let \(x=-1\), then \(f^{\prime\prime}(-1)=-30(-1)^{3}(-1 - 8)=-30\times(-1)\times(-9)=-270\lt0\). The function is concave - downward on \((-\infty,0)\).
• For \(0\lt x\lt8\), let \(x = 1\), then \(f^{\prime\prime}(1)=-30\times1^{3}(1 - 8)=-30\times(-7)=210\gt0\). The function is concave - upward on \((0,8)\).
• For \(x\gt8\), let \(x = 9\), then \(f^{\prime\prime}(9)=-30\times9^{3}(9 - 8)=-30\times729\times1=-21870\lt0\). The function is concave - downward on \((8,\infty)\).

Step3: Find inflection points

Since the concavity changes at \(x = 0\) and \(x = 8\), these are the \(x\) - coordinates of the inflection points.

Answer:

For concave upward: \((0,8)\)
For concave downward: \((-\infty,0)\cup(8,\infty)\)
For inflection points: \(x = 0,x = 8\)