Question

Verify mean value theorem f (x) =x3, x€(3, 4)

Answer 1

Answer:

The given function is f(x)=x

3

−5x

2

−3x

f being a polynomial function, so it is continuous in [1,3] and is differentiable in (1,3) whose derivative is 3x

2

−10x−3.

f(1)=1

3

−5⋅1

2

−3⋅1=−7,f(3)=3

3

−5⋅3

2

−3⋅3=−27

∴

b−a

f(b)−f(a)

=

3−1

f(3)−f(1)

=

3−1

−27−(−7)

=−10

Mean Value Theorem states that there exist a point c∈(1,3) such that f

′

(c)=−10

⇒3c

2

−10c−3=−10

⇒3c

2

−10c+7=0

⇒3c

2

−3c−7c+7=0

⇒3c(c−1)−7(c−1)=0

⇒(c−1)(3c−7)=0

⇒c=1,

3

7

, where c=

3

7

∈(1,3)

Hence, Mean Value Theorem is verified for the given function and c=

3

7

∈(1,3) is the point for which f

′

(c)=0