Question

what is the point of inflection of the function $y = 3(x - 2)^3+6$?
option #1: $(2, - 6)$
option #2: $(2,6)$
option #3: $(-2, - 6)$
(1 point)
the point of inflection is option #
check answer remaining attempts : 3

Explanation:

Step1: Find the first derivative

Use the chain rule. If \(y = 3(x - 2)^{3}+6\), then \(y^\prime=3\times3(x - 2)^{2}=9(x - 2)^{2}\)

Step2: Find the second derivative

Differentiate \(y^\prime = 9(x - 2)^{2}\) using the chain rule. \(y^{\prime\prime}=9\times2(x - 2)=18(x - 2)\)

Step3: Set the second derivative equal to zero

Set \(y^{\prime\prime}=0\), so \(18(x - 2)=0\). Solving for \(x\) gives \(x = 2\)

Step4: Find the \(y\) - value

Substitute \(x = 2\) into the original function \(y = 3(x - 2)^{3}+6\). \(y=3(2 - 2)^{3}+6=6\)

Answer:

Option #2: \((2,6)\)