Given as f(x) = x2 – 2x + 4 on [1, 5]
The every polynomial function is continuous everywhere on (−∞, ∞) and differentiable for all arguments. Here, f(x) is a polynomial function. Therefore it is continuous in [1, 5] and differentiable in (1, 5). Therefore both the necessary conditions of Lagrange’* mean value theorem is satisfied.
So, there exist a point c ∈ (1, 5) such that:
f(x) = x2 – 2x + 4
Differentiate with respect to x:
f’(x) = 2x – 2
For the f’(c), put the value of x=c in f’(x):
f’(c)= 2c – 2
For the f(5), put the value of x=5 in f(x):
f (5) = (5)2 – 2(5) + 4
= 25 – 10 + 4
= 19
For the f(1), put the value of x = 1 in f(x)
f (1) = (1)2 – 2 (1) + 4
= 1 – 2 + 4
= 3
Thus, lagrange'* mean value theorem is verified.
