Verify LMVT for the given functions f (x) = x² - 3x - 1, x ∈ [- 11/7 ,13/7]
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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.
(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.
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