Let and be two real-valued functions differentiable on an open interval .

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Let

f

and

g

be two real-valued functions differentiable on an open interval

I

. (i) Define an increasing function on an interval. (ii) What is meant by saying that “

f

grows faster than

g

” on

I

? (iii) Suppose there exists

c\in I

such that

f(c)=g(c)

and

f'(x)>g'(x)

for every

x\in I

. Give a clear verbal argument, without performing explicit calculations, to show that

f(x)>g(x)

for all

x>c

lying in

I

. What would you conclude for

x<c