Question
VERIFY ROLLES THEOREM f(x) = 9x³ - 4x in [ -2/3 , 2/3]
Answer
given f(x) = 9x³ - 4x
i) Continuity :-
since f(x) is a polynomial
therefore, it is continuous in [ -2/3 , 2/3]
ii) Differentiability :-
we have
f(x) = 9x³ - 4x
diff w.r.t "x" on both sides
f′(x) = 27x² - 4 -------->(1)
f′(x) is finite and unique ∀ x ∈ (-2/3 , 2/3)
therefore f′(x) exists
therefore f(x) is Differentiable in (-2/3, 2/3)
iii) f(a) = f(b) :-
we have
f(x) = 9x³ - 4x
f(-2/3) = 9(-2/3)³ - 4(-2/3)
f(-2/3) = -8/3 + 8/3
f(-2/3) = 0
f(2/3)
f(2/3) = 9(2/3)³ - 4(-2/3)
f(2/3) = 8/3 - 8/3
f(2/3) = 0
f(-2/3) = f(2/3)
therefore f(x) satisfies all the conditions of ROLLES THEOREM
verification :-
from equation 1
f′(x) = 27x² - 4
let f′(c) = 27c² - 4
27c² = 4
c² = 4/27
c = ± 2/3√3 ∈ (-2/3, 2/3)
therefore ROLLES THEOREM is verified
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