MathDB

Answer the following questions:
(1) Let

f(x)

be a function such that

f''(x)

is continuous and

f'(a)=f'(b)=0

for some

a<b

.
Prove that

f(b)-f(a)=\int_a^b \left(\frac{a+b}{2}-x\right)f''(x)dx

.
(2) Consider the running a car on straight road. After a car which is at standstill at a traffic light started at time 0, it stopped again at the next traffic light apart a distance

L

at time

T

. During the period, prove that there is an instant　for which the absolute value of the acceleration of the car is more than or equal to

\frac{4L}{T^2}.

Problem Statement