Question

Verify L.M.V.T. for the function f (x) = x (x+ 4)², x ∈ [0,4]

Answer 1

We have ,  f(x) = (x+1)/x  where ,  x ε [1, 3] .

(1) f(x) is a polynomial function , hence continuous in the interval [1, 3] .

(2) f(x) is a polynomial function , hence differentiable in the interval (1, 3) .

(3) f(1) =  1+1/1=2 , f(3) =3+1/3=4/3 .

Also ,  f’(x) = 1/x-(x+1)/x^2=-1/x^2

f'(c)=-1/c^2

Now , f’(c) = [f(b) – f(a)]/(b – a)

Or ,  -1/c^2= [f(3) – f(1)]/(3 – 1) = (4/3 – 2)/(2) = (-2/3)/2 = -1/3

Or , -1/c^2= -1/3 => c^ = 3,c=root3 ε (1, 3) .

Hence , Lagrange’s mean value theorem is verified.

Answer 2

first putting 0 in question
0 (0+4)^
= 4^
=16

now,
putting 4 in question
4 (4+4)^
=4 (8)^
= 4 (64)
=68