Question

1. (a) Use the Mean Value Theorem to establish the following inequality

e a (x − a) < ex − e a < ex (x − a), if a < x.

Answer 1

1.(a)

e ^a (x-a) < e^x - e ^a < e ^x (x-a) if a<x

Let f(x) = e^x

f'(x) = e^x

From Mean Value Theorem,

$\dfrac{f(b) - f(a)}{ (b-a)} = f'(c)$ ∀ c ∈ (a,b)

let a < x < b

so

$\dfrac{e^x - e^ a }{(x-a)} = e ^c$ .............................(1)

a < c < x

e ^a < e ^c < e^x

e ^a (x-a) < e^x - e ^a < e ^x (x-a) ..................from equation (1)