Question

3) verify MVT for fext the function. a) f(x) = 2x ^ 3 - 3x ^ 2 - x on [1, 2]​

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 exists 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)=0Step-by-step explanation:

Step-by-step explanation: