Letfandgbedifferentiablefunctionson(a,b).Supposethereisapointx0in(a,b)withf(x0)=g(x0)andf(x)

Question

Let f and g be differentiable functions on (a,b).Suppose there is a point x0 in (a,b) with
f (x0) = g(x0) and f (x)

Answer 1

设F(x)=g(x) - f(x).
因为在区间(a,b)上有f (x)<=g(x).所以在区间(a,b)上有F(x)>=0.
又已知存在a极点处导数为零.所以F'(x0) = g'(x0) - f '(x0)=0
可得 f '(x0) = g'(x0).