Find the greatest and least values

[utkarshakash]

Homework Statement

Find the greatest and least values of the function $f(x)=(sin^{-1}x)^3 +(cos^{-1}x)^3$

Homework Equations

The Attempt at a Solution

Setting f'(x)=0 and solving I get $|sin^{-1}x|=|cos^{-1}x|$

 

[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.

 

[Ray Vickson]

Homework Statement

Find the greatest and least values of the function $f(x)=(sin^{-1}x)^3 +(cos^{-1}x)^3$

Homework Equations

The Attempt at a Solution

Setting f'(x)=0 and solving I get $|sin^{-1}x|=|cos^{-1}x|$

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?

 

[utkarshakash]

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?

I mean arcsin x.

 

[LCKurtz]

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

 

[utkarshakash]

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

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

 

[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?

 

[LCKurtz]

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

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?

 

[utkarshakash]

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?

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