Question

what is the inflection point of $f(x)=(x + 7)^3-3$? (1 point)
$(-7,3)$
$(7,3)$
$(7,-3)$
$(-7,-3)$

Explanation:

Step1: Find the second - derivative

First, find the first - derivative using the chain rule. If \(y=(x + 7)^{3}-3\), then \(y^\prime=3(x + 7)^{2}\times1=3(x + 7)^{2}\).
Then, find the second - derivative. Using the chain rule again, \(y^{\prime\prime}=6(x + 7)\).

Step2: Set the second - derivative equal to zero

Set \(y^{\prime\prime}=0\), so \(6(x + 7)=0\).
Solve for \(x\): \(x+7 = 0\), which gives \(x=-7\).

Step3: Find the \(y\) - value

Substitute \(x =-7\) into the original function \(y=(x + 7)^{3}-3\).
\(y=(-7 + 7)^{3}-3=0-3=-3\).

Answer:

\((-7,-3)\)