Q. 2 F5.0( 1 Vote )

# Verify Rolle’s theorem for each of the following functions on the indicated intervals :

f(x) = x(x – 4)^{2} on [0, 4]

Answer :

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) = x(x – 4)^{2} on [0,4]

Since, given function f is a polynomial it is continuous and differentiable everywhere i.e., on R.

Let us find the values at extremums:

⇒ f(0) = 0(0 – 4)^{2}

⇒ f(0) = 0

⇒ f(4) = 4(4 – 4)^{2}

⇒ f(4) = 4(0)^{2}

⇒ f(4) = 0

∴ f(0) = f(4), Rolle’s theorem applicable for function ‘f’ on [0,4].

Let’s find the derivative of f(x):

⇒

Differentiating using UV rule,

⇒

⇒ f’(x) = ((x – 4)^{2}×1) + (x×2×(x – 4))

⇒ f’(x) = (x – 4)^{2} + 2(x^{2} – 4x)

⇒ f’(x) = x^{2} – 8x + 16 + 2x^{2} – 8x

⇒ f’(x) = 3x^{2} – 16x + 16

We have f’(c) = 0 cϵ(0,4), from the definition given above.

⇒ f’(c) = 0

⇒ 3c^{2} – 16c + 16 = 0

⇒

⇒

⇒

⇒

⇒ ϵ(0,4)

∴ Rolle’s theorem is verified.

Rate this question :

Verify the Rolle’s theorem for each of the functions

f(x) = x (x – 1)^{2} in [0, 1].

The value of c in Rolle’s theorem for the function f(x) = x^{3} – 3x in the interval

For the function the value of c for mean value theorem is

Mathematics - ExemplarVerify the Rolle’s theorem for each of the functions

Mathematics - Exemplar

Verify the Rolle’s theorem for each of the functions

f(x) = log (x^{2} + 2) – log3 in [– 1, 1].

Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :

f(x) = 3 + (x – 2)^{2/3} on [1, 3]

Using Rolle’s theorem, find the point on the curve y = x(x – 4), where the tangent is parallel to x-axis.

Mathematics - ExemplarVerify the Rolle’s theorem for each of the functions

Mathematics - Exemplar

State True or False for the statements

Rolle’s theorem is applicable for the function f(x) = |x – 1| in [0, 2].

Mathematics - ExemplarDiscuss the applicability of Rolle’s theorem on the function given by

Mathematics - Exemplar