VERIFY ROLLES THEOREM f(x) = 9x³ - 4x in [ -2/3 , 2/3]

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