Question

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

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 \(x\):

$$
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$$

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 \((-6,432)\)