<s>[INST] Find all numbers $c$ that satisfy Mean Value Theorem Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers $c$ that satisfy the conclusion of the Mean Value Theorem.
My function is a simple one: $x^{1/3}$ on the interval $[0,1]$.
Where I'm screwing up (I believe) is setting the derivative $1/3(x^{-2/3})$ equal to $1$.
Sorry if that too many parentheses or too silly of a question. It's my first on here.
I know (via the back of the book) that the answer should be $\frac{\sqrt{3}}{9}$
Thanks.
 [/INST] $$\dfrac{1}{3}x^{-2/3} = 1 \\ \implies \dfrac{1}{3}=x^{2/3} \\ \implies \left(\dfrac{1}{3}\right)^{3}=x^2 \\ \implies \sqrt{x^2}= \pm \sqrt{ \left(\dfrac{1}{3}\right)^{3}}$$ And for what it's worth, $$\sqrt{ \left(\dfrac{1}{3}\right)^{3}} = \sqrt{ \left(\dfrac{1}{3}\right)^{2}\left(\dfrac{1}{3}\right)} \\ = \left(\dfrac{1}{3}\right)\frac{1}{\sqrt{3}} \\ =  \left(\dfrac{1}{3}\right)\frac{1}{\sqrt{3}}  \cdot \frac{\sqrt{3}}{\sqrt{3}} \\ = \left(\dfrac{1}{3}\right)\frac{1}{3} \cdot \sqrt{3} \\ = \frac{\sqrt{3}}{9}$$
</s>