Question

Verify the Lagrange’s mean value theorem for the function: 
`f(x)=x + 1/x ` in the interval [1, 3]

Answer 1

`f(x)=x + 1/x `  x ∈ [1, 3] 
(i)  f (x) is continuous for    x ∈ [1,3] 
(ii)  f (x) is differentiable for  x ∈ (1,3) 
∴ LMVT is applicable. 
`f(1)=1+1/1 = 2 and  f(3)= 3 + 1/3 = 10/3 `

`f'(x)= 1- 1/x^2     therefore f'(c)= 1-1/c^2`

`therefore f'(c)= (f(b)-f(a))/(b - a) rArr 1-(1)/c^2= (10/3 -2)/(3-1) = ((10-6)/3)/2 = 4/6=2/3`

`rArr 1- 2/3 = 1/c^2 rArr 1/3=1/c^2 ∴ c^2 = 3`

∴ c = ± `sqrt3   but  c=sqrt3 ∈ [1,3]`
LMVT is vertified.