In proving Cauchy's mean value theorem, the first step is to use this function:
$$ h(x)=[f(b)−f(a)]g(x)−[g(b)−g(a)]f(x)$$
I've seen this in many textbooks but none of them actually show how they got this function. I understand every steps in the proof of Cauchy's mean value theorem, except how to derive this equation myself.
Do you guys know how do they get this function? Thanks a lot!
Best Answer
To prove the theorem, since the fractions create problems for derivatives, cross multiply. Then you need to show that
$$[f(b)-f(a)]g'(x)-[g(b)-g(a)]f'(x)=0 \, \mbox{for some} \, x$$
Now, the left hand side is the derivative of some function, which is that function?