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Examine the applicability of Mean Value Theorem for all the three functions being given below: [Note for students: Check exercise 2] (i) f(x)=x^{2}-1 for x \in[1,2]
CBSE | Class 12 | Continuity and Differentiability | Exercise 5.8 | Maths | NCERT
Examine the applicability of Mean Value Theorem for all the three functions being given below: [Note for students: Check exercise 2] (i) f(x)=x^{2}-1 for x \in[1,2]

Solution:

(i) f(x)=x^{2}-1 for x \in[1,2]……….(1)

Provided here, f(x) is a polynomial function.

Then, f(x) is continuous and derivable on the real line.

Therefore f(x) is continuous in the closed interval [1,2] and derivable in open interval (1,2).

As a result, both conditions of M.V.T. are satisfied.

From the equation (1), we have

f^{\prime}(x)=2 x

f^{\prime}(c)=2 c

Now again, from the equation (1):

f(a)=f(1)=(1)^{2}-1=1-1=0

And, f(b)=f(2)=(2)^{2}-1=4-1=3

Then,

f^{\prime} c=\frac{f(b)-f(a)}{b-1}

2 c=\frac{3-0}{2-1}

c=\frac{3}{2} \in(1,2)

As a result, Mean Value Theorem is verified.

i

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