##### ANSWERS to exercise 3.8

2. (a) Increases ∀*x* ∈ *R*

(b) Increases in (−1, 0) and decreases in (0, 1)

(c) Increases ∀*x* ∈ *R*

6. Decreases

##### 3.5 Generalised Mean Value Theorem

In many applications it is useful to approximate a continuous function by a polynomial function which is the simplest continuous function. Taylor’s and Maclaurin’s theorems are important tools which provide such an approximation for real functions. Mean value theorems relate the value of the functions and its first order derivative, where as, Taylor’s and Maclaurin’s theorems generalise this relation to higher order derivatives. Hence these theorems can be considered as “*Generalised mean value theorems”.*

#### 3.5.1 Taylor’s Theorem with Lagrange’s form of Remainder

If *f* is a real function ...