Suppose is continuous on and differentiable on

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Suppose

f

is continuous on

[a,b]

and differentiable on

(a,b)

. Assume that there exists a constant

k>0

such that

f'(x)\ge k

for all

x

(a,b)

. (1) Show that for every

x

and

y

[a,b]

with

x<y

, the inequality

f(y)-f(x)\ge k(y-x)

holds. (2) Deduce that

f

is strictly increasing on

[a,b]

and, consequently, that the equation

f(x)=c

can have at most one solution in that interval for any real constant

c