Question

find the x and y coordinates of all inflection points. f(x)=x^3 + 15x^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

Given \(f(x)=x^{3}+15x^{2}\), using the power rule \((x^n)^\prime = nx^{n - 1}\), we have \(f^\prime(x)=3x^{2}+30x\).

Step2: Find the second - derivative

Differentiate \(f^\prime(x)\) with the power rule. \(f^{\prime\prime}(x)=(3x^{2}+30x)^\prime=6x + 30\).

Step3: Set the second - derivative equal to zero

To find the potential inflection points, set \(f^{\prime\prime}(x)=0\). So, \(6x+30 = 0\).

Step4: Solve for \(x\)

Subtract 30 from both sides: \(6x=-30\), then divide by 6, we get \(x=- 5\).

Step5: Find the \(y\) - coordinate

Substitute \(x = - 5\) into the original function \(f(x)\). \(f(-5)=(-5)^{3}+15\times(-5)^{2}=-125 + 375=250\).

Answer:

A. The inflection point(s) is/are \((-5,250)\)