Question

f(x) = 4x^3 + 15x^2 - 150^x + 4
(a) Find the intervals on which f is increasing. (Enter the interval that contains smaller numbers first.)


Find the interval on which f is decreasing.


(b) Find the local minimum and maximum values of f.


(c) Find the inflection point.


Find the interval on which f is concave up.


Find the interval on which f is concave down.

Answer 1

so, take your equation and solve for the first derivative (2x^2 +30x-150). set that to 0 and solve for 0's. These are your critical values. Draw a number line and test for values in between your zeroes. Let's say if 1 is one of your zeroes and so is 5, test say 2, or 3, and see if you get a positive or negative number. Wherever you get positive values in between zeroes for f', the function is increasing. Okay, to find the min you need a zero that goes from - values to +. For maximum, the values need to go from + to - in f'. For concavity and inflection points you need to do the same type of thing with the second derivative f'' =4x+30. Set that to 0 to get another critical value, and test what the values do before and after. If they go from + to - or vice versa, then the critical value is an inflection point. The interval in which the f'' is +, f is concave upward, if - then it is concave downward. These are the first and second derivative tests for critical values.