Joe Vaskas

Math Question of the Month (if you wish, send email comments to jav415@gmail.com):

Let f(x) = X3 -3x2 -9x +10. Find the intervals where f(x) is increasing and decreasing. Find all relative extrema.  

Solution: 

The evaluation of the derivative of a function at a point represents the slope of a tangent line to the function at that point. As such, if you find the points were the derivative = 0 (slope is 0; the tangent line is parallel to the x axis) , you have identified where the curve changes direction. That is, the relative maximum and minimum points.

1. Find the derivative, f'(x),  of the function. f’(x) = 3x2- 6x - 9.

2. Set the derivative = 0. 

3x2- 6x - 9 = 0, now factor

(X + 1)(x – 3) =0

X = -1, x = 3  So the graph turns at x= -1 and x= 3

3.  Evaluate (a) to the left of -1,  (b) the interval between x= -1 and x= 3 and finally (c) to the right of x= 3 by inserting for example,  -2, 0, and 4 respectively into the equation of the derivative to determine the slope at those points.

4.

·   for the interval (-infinity, -1),  f’(x) is greater than 0, therefore the function is increasing (slope is positive)

·   for the interval (-1, 3), f’(x) is less than 0, therefore the function is decreasing (slope is negative)

·   for the interval (3, infinity), f’(x) is greater than zero, therefore the function is increasing

5. Therefore, there is a relative maximum at x= -1 because the slope changes from positive to negative. Also, there is a relative minimum at x= 3 because the slope changes here from negative to positive.