Question

If f and g are differentiable functions on the interval [a,b] with f(a)=g(a) and f(b)=g(b), prove that at some point in the interval [a,b], f and g have parallel tangent lines

And then prove that the result from above still holds if the assumptions f(a)=g(a) and f(b)=g(b) are relaxed to requiring f(b)-f(a)=g(b)-g(a)

Answer 1

Consider the function h(x) = f(x) - g(x). Then apply the MVT with end points a and b.

Answer 2

I'm not sure how to use that

Answer 3

Well h(a) = h(b) = 0. Hence by the MVT, there exists an x in the interval (a,b) such that
\[ h'(x) = \frac{ h(b) - h(a) }{b-a} = 0 \]
hence ....

Answer 4

after that how do you prove that there are parallel tangents?

Answer 5

what does it mean for the tangents to the graphs of f and g to be parallel?

Answer 6

that the slope of a line equals 0 along a point of a particular function right?

Answer 7

no.

Answer 8

what is the relation between the slope of the tangent line to the graph of a function f, and the derivative f' ?

Answer 9

f' shows the slope of f if you graph it

Answer 10

f'(x) is the slope of the tangent line to the graph of f at x.

Answer 11

So if there is an x such that h'(x) = 0, then ....

Answer 12

it's the tangent line

Answer 13

i'm just so confused my professor sucks at teaching so i'm having problems grasping the idea

Answer 14

What is the equation of the tangent line to the graph of a function f at x = a?

Answer 15

y = .... what?

Answer 16

y=ax+b

Answer 17

relate it to the function f.

Answer 18

how?

Answer 19

y = ax + b is the general equation for a straight line. what must a and b be equal to in order that the line is the tangent to the graph of f at x = a?

Answer 20

i really am so confused

Answer 21

Ok. You need to go back to your text book and re-read carefully the section on the tangents to the graphs of functions.

THis is a basic level of understanding you need to have in order to answer this question.

The first answer I gave you will make very little sense without it.

After you've boned up on the equations of tangents, read the section on Rolle's Theorem and the Mean Value Theorem.

Then you'll have all the technical understanding you need to answer this question.

In the meantime, draw a diagram of two functions f and g such that f(a) = g(a) and f(b) = g(b) and convince yourself the result you're being asked to prove makes intuitive sense.

Answer 22

ok i appreciate your help..just going crazy cuz im mixing things up and confusing myself cuz i have so many proofs to do...lol