Question

please solve question no 11

Answer 1

Since polynomials are continuous and differentiable eveyrwhere, f(x) is continuous in [2,3] and differentiable in (2,3).
f(2) = 0 and f(3) = 0

As all the conditions of Rolle's theorem are satisfied, by Rolle's Theorem,
there exists c where c ∈ (2,3) such that
f' (c) = 0
⇒ -9(c-2)^4 (c-3)² - 8 (c-3)³(c-2)³ = 0
⇒ -9(c-2) = 8(c-3)
⇒ -9c + 18 = 8c - 24
⇒ 17c = 42
⇒ c = 2.47

Since c = 2.47 ∈ (2,3) , Rolle's Theorem is verified.