It is given that the Rolle’s theorem holds for the function f(x) = x3 + bx2 + cx, xϵ[1,2] at the point x = 4/3. Find the values of b and c.
First, let us write the conditions for the applicability of Rolle’s theorem:
For a Real valued function ‘f’:
a) The function ‘f’ needs to be continuous in the closed interval [a,b].
b) The function ‘f’ needs differentiable on the open interval (a,b).
c) f(a) = f(b)
Then there exists at least one c in the open interval (a,b) such that f’(c) = 0.
Given function is:
⇒ f(x) = x3 + bx2 + cx, xϵ[1,2]
According to the problem the Rolle’s theorem holds for the function ‘f’ at 
.
We can say that 
.
Let’s find the derivative of f(x)
⇒ 
⇒ 
⇒ f’(x) = 3x2 + 2bx + c
We have 
⇒ 
⇒ 
⇒ 
⇒ 8b + 3c = – 16 ...... (1)
We also have f(1) = f(2)
⇒ (1)3 + b(1)2 + c(1) = (2)3 + b(2)2 + c(2)
⇒ 1 + b(1) + c = 8 + b(4) + 2c
⇒ 3b + c = – 7 ......(2)
On solving (1) and (2), we get
⇒ b = – 5 and c = 8
∴ The values of b and c is – 5 and 8.