# Chapter 3 - Applications of Differentiation - 3.2 Exercises: 38

The Mean Value Theorem can be applied; $c=2\sqrt{3}.$

#### Work Step by Step

Since $f(x)$ is a polynomial, it is continuous for all values of $x$ and differentiable at every value of $x.$ $a=0; b=6\to f(a)=0$ and $f(b)=432$ Since $f(x)$ is continuous over $[a, b]$ and differentiable over $(a, b)$, applying Mean Value Theorem over the interval $[a , b]$ guarantees the existence of at least one value c such that $a\lt c\lt b$ and $f′(c)=\dfrac{f(b)-f(a)}{b-a}.$ $f'(c)=\dfrac{432-0}{6-0}=72.$ $f'(x)=6x^2\to 6x^2=72\to c=2\sqrt{3}$

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