Expert: Paul Klarreich - 4/28/2007

Question
Let f be a differentiable function of x that satisfies f(1)=7 and f(4)=3. which of the following conditions would guarantee that the tangent line at x=c is parallel to the secant line joining (1,f(1)) to (4,f(4)?
i tried to do this problem but i'm not sure where to start. but i treid to figure out. can you please show me all the steps so i can understand.
thanks

Answer
Questioner:   h
Category:  Calculus
 
Question:  Let f be a differentiable function of x that satisfies f(1)=7 and f(4)=3. which of the

following conditions would guarantee that the tangent line at x=c is parallel to the secant line

joining (1,f(1)) to (4,f(4)?
i tried to do this problem but i'm not sure where to start. but i tried to figure out. can you please show me all the steps so i can understand.
thanks
...................................
Hi, again, h,

This looks like the Mean Value Theorem, which says:

[VIEW THIS IN COURIER FONT]

If f(x) is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) where:
        f(b) - f(a)
f'(c) =  -----------
          b - a

Applying this to your example, you have:

a = 1
b = 4
f(a) = f(1) = 7
f(b) = f(4) = 3
Then the line through those two points has a slope equal to:
   f(4) - f(1)
m = -----------
     4  - 1

   3  -  7
 = -------
   4  - 1

   - 4
 = -------
     3

and the MVT says there exists  c in (1,4) where  f'(c), which is the slope of your tangent line, is equal to -4/3, which is the slope of your secant line.