VERIFY ROLLES THEOREM for f(x) = x² - 5x + 6 in [ 2,3] || #swm_media

 

Question :

 VERIFY ROLLES THEOREM f(x) = x² - 5x + 6 in [ 2,3] 

f(x) = x² - 5x + 6 in [2,3]

Answer :-

given f(x) = x² - 5x + 6

Continuity :-

since f(x) is a polynomial

therefore it is continuous in [2,3]

Differentiability :-

we have 

f(x) = x² - 5x + 6

differentiate w.r.t "x" on both sides 

f'(x) = 2x - 5  ---->equation (1)

f′(x) is finite and unique ∀ x ∈ (2,3)

therefore f'(x) exists

therefore is Differentiable in (2,3)

f(a) = f(b) :-

we have

f(x) = x² - 5x + 6

f(2) = 2² - 5(2) + 6

f(2) = 4 - 10 + 6

f(2) = 10 - 10

f(2) = 0

f(2) = 3² - 5(3) + 6

f(2) = 9 - 15 + 6

f(2) = 15 - 15

f(2) = 0

f(2) = f(3)

therefore f(x) satisfied by all the conditions of ROLLES THEOREM 

Verification :-

from equation (1)

f'(x) = 2x - 5 

let f'(c) = 2c - 5

      2c = 5

        c = 5/2

         c = 2.5 ∈ (2,3)

therefore ROLLES THEOREM is verified.