Find the values of $x$ for which the given function is concave up, the values of $x$ for which it is concave down, and any points of inflection. $y=-x^4+12x^3-12x+13$ . What is/are the point(s) of inflection of $y$ ? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. $\newline$ A. enter your response here (Type an ordered pair. Use a comma to separate answers as needed.) $\newline$ B. There are no points of inflection.

Full solution

Q. Find the values of

x

for which the given function is concave up, the values of

x

for which it is concave down, and any points of inflection.

y=-x^4+12x^3-12x+13

. What is/are the point(s) of inflection of

y

? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

\newline

A. enter your response here (Type an ordered pair. Use a comma to separate answers as needed.)

\newline

B. There are no points of inflection.

Calculate Second Derivative: Now, calculate the second derivative: $y'' = -12x^2 + 72x$ .
Find Points of Inflection: To find points of inflection, set the second derivative equal to zero and solve for $x$ : $-12x^2 + 72x = 0$ . Factor out $12x$ : $12x(-x + 6) = 0$ .
Solve for Potential Points: Solve for $x$ : $x = 0$ or $x = 6$ . These are the potential points of inflection.
Determine Concavity: To determine concavity, test intervals around the critical points $0$ and $6$ in the second derivative. Pick test points like $x = -1$ , $x = 3$ , and $x = 7$ and plug them into $y''$ .
Test Intervals in Second Derivative: For $x = -1$ : $y''(-1) = -12(-1)^2 + 72(-1) = -12 - 72 = -84$ , which is less than $0$ , so the function is concave down on $(-\infty, 0)$ .
Concave Down on $(-\infty, 0)$ : For $x = 3$ : $y''(3) = -12(3)^2 + 72(3) = -108 + 216 = 108$ , which is greater than $0$ , so the function is concave up on $(0, 6)$ .
Concave Up on $(0, 6)$ : For $x = 7$ : $y''(7) = -12(7)^2 + 72(7) = -588 + 504 = -84$ , which is less than $0$ , so the function is concave down on $(6, \infty)$ .
Concave Down on $(6, \infty):</b> To find the \$y$ -coordinates of the points of inflection, plug $x = 0$ and $x = 6$ into the original function $y$ . For $x = 0$ : $y(0) = -(0)^4 + 12(0)^3 - 12(0) + 13 = 13$ . For $x = 6$ : $y(6) = -(6)^4 + 12(6)^3 - 12(6) + 13 = -1296 + 2592 - 72 + 13 = 1237$ .