# Find the greatest and least values

1. Jun 10, 2013

### utkarshakash

1. The problem statement, all variables and given/known data
Find the greatest and least values of the function $f(x)=(sin^{-1}x)^3 +(cos^{-1}x)^3$

2. Relevant equations

3. The attempt at a solution
Setting f'(x)=0 and solving I get $|sin^{-1}x|=|cos^{-1}x|$

2. Jun 10, 2013

### verty

What does $sin^{-1}{x}$ mean? The remainder of the question is a test of your intuition. I hope no one gives too much help.

3. Jun 10, 2013

### Ray Vickson

The notation $\sin^{-1} x$ is equally likely to mean $1/ \sin\, x$ or $\arcsin\, x$. After all, the notation $\sin^n x$ is taken to mean $(\sin \, x)^n$ whenever $n \neq -1$! So, which do you mean?

4. Jun 10, 2013

### utkarshakash

I mean arcsin x.

5. Jun 10, 2013

### LCKurtz

So have you looked at the graphs of |arcsin(x)| and |arccos(x)|?

6. Jun 11, 2013

### utkarshakash

I get x= 0.707 by plotting the graph. But I need two values.

Last edited: Jun 11, 2013
7. Jun 11, 2013

### jeppetrost

Say, there is a point (x,y) where |asin(x)|=|acos(x)|=y, what does that say about sin(y) and cos(y)? Can you use that?

8. Jun 11, 2013

### LCKurtz

Can you use analysis to get the exact value?

For a function continuous on a closed interval, where can the possible maximum and minimum points occur? What is the domain in this problem?

9. Jun 12, 2013

### utkarshakash

I am thinking it the other way. I can rewrite the original expression as
$\pi /2 \left( \pi ^2 /4 - 3sin^{-1} x cos^{-1} x \right)$

Last edited: Jun 12, 2013