Determine whether the function
$$f(x)=\sin^{-1} x$$
satisfies the conditions of the Mean Value Theorem on the interval [0,1/2]. If so find the points guaranteed to exist by the theorem

Question

Determine whether the function 
$$f(x)=\sin^{-1} x$$
satisfies the conditions of the Mean Value Theorem on the interval [0,1/2].  If so find the points guaranteed to exist by the theorem

zehanz · Answer

Here is a graph of the situation (in red).
The dotted line is the line between the endpoints.

Conditions of the MVT:
1. contiuity on [0, 1/2]: OK
2. differentiability on (0, 1/2): OK

So the conditions are met.
Now look for a point on the graph where the tangent line has the same slope as the dotted line.

anonymous · Answer

that is a hard look.
you need to solve 
$$f'(x)=\frac{f(b)-f(a)}{b-a}$$ for $x$

anonymous · Answer

the right hand side is $\frac{\sin^{-1}(\frac{1}{2})-\sin^{-1}(0)}{\frac{1}{2}-0}$

anonymous · Answer

once you evaluate, you should get 
$$\frac{\pi}{6}\times 2=\frac{\pi}{3}$$

anonymous · Answer

oh I see thanks guys

anonymous · Answer

you still have to take the derivative, set it equal to $\frac{\pi}{3}$ and solve

anonymous · Answer

yea I know that but i got the wrong answer for the first part now that I see where the mistake was made the next part is easy $$\sqrt{1-\frac{ 9 }{ \pi ^{2} }}$$

zehanz · Answer

@satellite73: Although the graph of $sin^{-1}x$ is almost straight, the dotted line is still visible and helps solving the problem imo  :)