find the x and y coordinates of all inflection points.\nf(x)=x^{3}+18x^{2}\nwhat is/are the inflection…

find the x and y coordinates of all inflection points.\nf(x)=x^{3}+18x^{2}\nwhat is/are the inflection point(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the inflection point(s) is/are\n(type an ordered pair. use a comma to separate answers as needed.)\nb. there are no inflection points.

find the x and y coordinates of all inflection points.\nf(x)=x^{3}+18x^{2}\nwhat is/are the inflection point(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the inflection point(s) is/are\n(type an ordered pair. use a comma to separate answers as needed.)\nb. there are no inflection points.

Answer

Explanation:

Step1: Find the first derivative

Using the power rule ((x^n)^\prime = nx^{n - 1}), for (y = f(x)=x^{3}+18x^{2}), we have (f^\prime(x)=3x^{2}+36x).

Step2: Find the second derivative

Differentiate (f^\prime(x)) again. Using the power rule, (f^{\prime\prime}(x)=(3x^{2}+36x)^\prime = 6x + 36).

Step3: Set the second derivative equal to zero

Set (f^{\prime\prime}(x)=0), so (6x+36 = 0). Solving for (x): [ \begin{align*} 6x&=- 36\ x&=-6 \end{align*} ]

Step4: Find the (y) - coordinate

Substitute (x = - 6) into the original function (y=f(x)). (f(-6)=(-6)^{3}+18\times(-6)^{2}=-216 + 648=432).

Answer:

A. The inflection point(s) is/are ((-6,432))