Question

State the values of x for which f is decreasing. Use +infinity or -infinity for the symbols. f(x) = x^4 - 40 x^3 and State the values of x for which f is concave down. here's what i have found: the values of x when slope = 0 is (0,30) maximum and minimum are: none and 30. one of the inflection points is (0,?). i know the first part should look like when f is decreasing : -infinity 14 years ago

Answer 1

Taking the first derivative one gets 4x^3 -120 x^2. There are relative extrema at x=0 and x=30.

Plugging a value less that 0 into the derivative equation gives us a positive value. Therefore, the function is increasing on (-infinity, 0). Plugging in a value between 0 and 30 gives us a negative, therefore it is decreasing on (0,30). Plugging in a value greater than 30 gives us a positive, therefore it is increasing on (30, infinity)

Answer 2

For the inflection point find the second derivative. In this case it is 12x^2 -240x. There are inflection points at x=0 and x=20.

Plugging a negative value gives us a positive answer. This means it is concave up on (-infinity, 0). A value between 0 and 20 gives us a negative answer, so it is concave down on (0,20), and a value greater than 20 is positive, so it is concave up on (20, infinity)