A function is defined as for in the interval

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A function is defined as

f(x)=k\bigl(\tan x-1\bigr)+3

for

x

in the interval

\left[0,\dfrac{\pi}{4}\right]

. The average value of

f

over this interval is known to be

4

. (i) Obtain the equation that must be satisfied by

k

and hence determine the value of

k

. (ii) For the constant

k

found in part (i), the Mean Value Theorem for Integrals guarantees a point

c

\left(0,\dfrac{\pi}{4}\right)

such that

f(c)=4

. Find the exact value of

c