96.3k views
2 votes
Find all relative extrema of the function. use the second derivative test where applicable. (if an answer does not exist, enter dne.) f(x) = x3 − 9x2 + 2

User Md Masud
by
8.7k points

1 Answer

4 votes
First find the derivative of the function. The derivative is
f'(x)=3 x^(2) -18x. Now set it equal to 0 to find the critical numbers.
0=3 x^(2) -18x. Factor to solve for the zeros of the derivative.
0=3x(x-6). So 3x = 0, and x = 0, or x - 6 = 0 and x = 6. We will make a table with values for -∞<x<0, 0<x<6, 6<x<∞. Pick a value within those boundaries for each interval and find the sign, positive or negative, that results from subbing that number into the derivative. If we choose -1 in the first interval f'(-1)=21, so the function is increasing from negative infinity to 0. If we choose 1 in the second interval, f'(1)=-15, so the function is decreasing from 0 to 6. If we choose 10 in the last interval, f'(10)=120, so the function is increasing from 6 to infinity. The points of extrema are found by subbing the critical x values into the original function. We know the function is increasing from negative infinity to 0, so f(0)=2, and our max point is (0, 2). We know the function is decreasing from 6 to infinity, so f(6)=-106, and our min point is (6, -106). I do this instead of the second derivative test, but they both work.
User Todd Curry
by
8.4k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories