Question

If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such that f '(c) = f(b) − f(a)b − a. (Enter your answers as a comma-separated list. If the Mean Value Theorem cannot be applied, enter NA.)

Answer 1

We need to satisfy the following conditions:

1. The function is continuous on the closed interval [a,b]

2. f is differentiable on the open interval (a,b).

Let's check every condition:

`\begin{gathered} f(x)=4x^3 \\ \lbrack1,2\rbrack \end{gathered}`

The function is continuous since there are no any restrictions.

`f^(\prime)(x)=12x^2`

The derivative of the function exists over the interval (1,2)

Therefore:

Answer:

Yes, The mean value theorem can be applied

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`\begin{gathered} f^(\prime)(c)=(f(b)-f(a))/(b-a) \\ f(b)=f(2)=4(2)^3=32 \\ f(a)=f(1)=4(1)^3=4 \\ so\colon \\ (32-4)/(2-1)=28 \end{gathered}`
`\begin{gathered} f^(\prime)(c)=28 \\ 28=12x^2 \\ x^2=(28)/(12) \\ x=\sqrt[]{(7)/(3)} \\ \end{gathered}`

Therefore: