Q. 74.5( 2 Votes )

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

Answer :

Condition (1):

Since, f(x)=x^{3}- 7x^{2}+16x-12 is a polynomial and we know every polynomial function is continuous for all xϵR.

⇒ f(x)= x^{3}- 7x^{2}+16x-12 is continuous on [2,3].

Condition (2):

Here, f’(x)=3x^{2}-14x+16 which exist in [2,3].

So, f(x)= x^{3}- 7x^{2}+16x-12 is differentiable on (2,3).

Condition (3):

Here, f(2)= 2^{3}- 7(2)^{2}+16(2)-12=0

And f(3)= 3^{3}- 7(3)^{2}+16(3)-12=0

i.e. f(2)=f(3)

Conditions of Rolle’s theorem are satisfied.

Hence, there exist at least one cϵ(2,3) such that f’(c)=0

i.e. 3c^{2}-14c+16=0

i.e. (c-2)(3c-7)=0

i.e. c=2 or c=7÷3

Value of

Thus, Rolle’s theorem is satisfied.

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].

Verify the Rolle’s theorem for each of the functions

f(x) = x(x + 3)e^{-x/2} in [–3, 0].

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