What is the inflection point of \( f(x)=(x+7)^{3}-3 \) ? (1 point) \( (7,3) \) \( (-7,3) \) \( (7,-3) \) \( (-7,-3) \)

To find the inflection point of the function \( f(x) = (x+7)^3 - 3 \), we first find the second derivative. The first derivative is \( f'(x) = 3(x+7)^2 \), and the second derivative is \( f''(x) = 6(x+7) \). Setting the second derivative to zero gives \( 6(x+7)=0 \Rightarrow x = -7 \). Substituting \( x = -7 \) back into the original function gives \( f(-7) = (-7+7)^{3} - 3 = -3 \). Thus, the inflection point is \( (-7,-3) \). That's the mathematical side, but did you know that inflection points can be super helpful in real-life applications? For instance, in economics, understanding where a cost or revenue function shifts can inform businesses about optimal pricing strategies or production levels! And speaking of math, if you're ever unsure about finding inflection points, remember to check both the first and second derivatives. A common mistake is to only consider the first derivative. Make sure you find the second derivative and confirm the sign change—this guarantees that you've indeed found an inflection point!