Increasing and decreasing functions in business calculus

Question

In this problem, I do not comprehend how the fraction x=33/23 is arrived at for the critical number. I am told just to divide -6.6 by -4.6 to get x.

The problem is; For the function F(x) = 3.1+6.6x-2.3x² 

Find a. the critical numbers b. The open intervals where the function is increasing. and c. The open intervals where the function is decreasing.

Thank you for any insight you may provide.

Answer 1

The critical points are the values of x such that f '(x)=0.

 

f(x) = -2.3x² + 6.6x + 3.1

 

f '(x) = -4.6x + 6.6

The critical point(s) occurs when f '(x) = 0:

0 = -4.6x + 6.6

4.6x = 6.6

x = 6.6/4.6

 

You don't want a fraction with a decimal in it, so multiply 6.6/4.6 by 5/5:

 

x = (6.6/4.6)*(5/5) = 33/23

 

To determine if this is a maximum or minimum, take f''(33/23).  If it's > 0, it's a minimum.  If it's < 0 it's a maximum.  Since there is only one critical point, there is one decreasing interval and one increasing interval. that changes at x = 33/23.

Answer 2

The critical values are when the derivative = - of the derivative is undefined.

 

So your derivative is f'(x) = 6.6 - 4.6x 

This is just using the polynomial rule for taking derivatives.

 

So the critical values are 6.6-4.6x = 0 or x= -6.6/-4.6 = 33/23

 

So on a number line, the value of the derivative is 0 at 33/23.

If I take a number less than 33/23....let's say 1...and plug it into the derivative, I get f'(1)=6.6-4.6(1) = 2

Since it is a positive number, the function is increasing.

If I plug in a number greater than 33/23....let's say 2....f'(2)= 6.6-4.6(2) = -2.6

since it is negative, the function is decreasing.

 

So your function increases from (-infinity,33/23) and it decreases from (33/23, infinity)

 

Hope this helped