find the x and y coordinates of all inflection points.\nf(x)=x³ + 27x²\nwhat is/are the inflection point(s)…

find the x and y coordinates of all inflection points.\nf(x)=x³ + 27x²\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)=x³ + 27x²\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

Use the power rule ((x^n)^\prime=nx^{n - 1}). If (f(x)=x^{3}+27x^{2}), then (f^\prime(x)=3x^{2}+54x).

Step2: Find the second - derivative

Differentiate (f^\prime(x)) with respect to (x). (f^{\prime\prime}(x)=(3x^{2}+54x)^\prime = 6x + 54).

Step3: Set the second - derivative equal to zero

Solve the equation (f^{\prime\prime}(x)=0). (6x+54 = 0). Subtract 54 from both sides: (6x=-54). Divide both sides by 6: (x=-9).

Step4: Find the (y) - coordinate

Substitute (x = - 9) into the original function (f(x)). (f(-9)=(-9)^{3}+27(-9)^{2}). (f(-9)=-729 + 27\times81). (f(-9)=-729+2187). (f(-9)=1458).

Answer:

A. ((-9,1458))