Question

f(x)=x+1/x x ∈ [1,3],Apply Mean theorem and Find C.

Answer 1

HOPE YOU ARE UNDERSTAND

Answer 2

Mean value theorem : Let f(x) is the continuous function in closed interval [a, b] and differentiable function in open interval (a, b).

then according to mean value theorem, there must be a point c between a and b such that f'(c) = {f(b) - f(a)}/(b - a)

here, f(x) = x + 1/x , x ∈ [1, 3]

so, f'(x) = 1 + (-1/x²) = 1 - 1/x²

let a point c ∈ (1, 3) such that, f'(c) = (f(3) - f(1))/(3 - 1)

or, 1 - 1/c² = [3 + 1/3 - 1 - 1/1]/2

or, 1 - 1/c² = [2 - 2/3 ]/2

or, 1 - 1/c² = 2/3

or, 1 - 2/3 = 1/c²

or, c² = 3 ⇒c = ±√3

but √3 ∈ (1, 3)

hence, c = √3