find the x and y coordinates of all inflection points. f(x)=x^3 + 15x^2. what is/are the inflection…
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.
Answer
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))