Mathematics

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

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

k

and hence determine the value of

k

. b. (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

Intuitive explanation

The average value of a function over an interval is the integral of that function divided by the interval's length—think of it as the "height of a rectangle with the same area." Here, you'll set up the average value formula, integrate $f(x)$ over $[0, \frac{\pi}{4}]$ , and use the condition that this average equals 4 to solve for the unknown parameter $k$ . The key is recognizing that $\tan x$ integrates to $-\ln(\cos x)$ , and carefully evaluating the antiderivative at both endpoints.

Intuitive explanation

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