How do you graph #f(x)=6x-15# using the information given by the first derivative?

Question

Ezra Adams · Accepted Answer

The function is of the form #y=mx+c# which from early secondary school mathematics should be recognised as that of a straight line with slope #m# and #y#-intercept #-15#.  You would not use, nor do you require,  Calculus to provide any further insight into this function!  graph{y=6x-15 [-40, 40, -20, 20]}

Charles Anctil · Answer

undefined To graph $ f(x) = 6x - 15 $ using the information given by the first derivative, follow these steps:

1. Find the first derivative $ f'(x) $.
   $ f'(x) = \frac{d}{dx}(6x - 15) $
   $ f'(x) = 6 $

2. Determine the critical points by setting $ f'(x) $ to zero.
   $ 6 = 0 $ has no solution, so there are no critical points.

3. Identify the intervals where $ f'(x) $ is positive or negative.
   $ f'(x) = 6 $ is always positive.

4. Determine the behavior of $ f(x) $ in the intervals.
   Since $ f'(x) $ is always positive, $ f(x) $ is always increasing.

5. Sketch the graph:
   - The graph of $ f(x) = 6x - 15 $ is a straight line with a slope of 6 and a y-intercept of -15.
   - The graph is always increasing and has no turning points or inflection points.

Thus, the graph of $ f(x) = 6x - 15 $ is a straight line that increases continuously without any critical points.