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.
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