Question

It is given that the Rolle’s theorem holds for the function f(x) = x³ + bx² + cx, x∈[1, 2] at the point x = 4/3. Find the values of b and c.

Answer 1

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, x∈[1, 2]

According to the problem the Rolle’s theorem holds for the function ‘f’ at x = \(\frac{4}{3}.\)

We can say that f'(\(\frac{4}{3}\)) = 0.

Let’s find the derivative of f(x)

⇒ 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.