Find the range of $\displaystyle \frac{1}{2bx - (x^2 + b^2 + \sin^2 x)}$, $\displaystyle x\in [-1,0], b\in [2,3]$.

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- May 23rd 2009, 10:11 PMfardeen_genFind range?
Find the range of $\displaystyle \frac{1}{2bx - (x^2 + b^2 + \sin^2 x)}$, $\displaystyle x\in [-1,0], b\in [2,3]$.

- May 24th 2009, 07:28 AMjosipive
Generaly:

Bol. - Weier. theorem says: if f is continuous on [ a, b ] then f( [ a, b ] ) = [ c, d ] segment. Candidates for min and max ( c and d ) are:

1. z = a and z = b ( f( a ) and f( b ) )

2. f is differential in z and f ' ( z ) = 0 ( f( z ) )

3. f is not differential in z - Jun 21st 2009, 12:42 PMfardeen_gen
The answer given is $\displaystyle R_f = \left[\frac{-1}{\sin^2 (1) + 9}, \frac{-1}{16}\right]$. How did we get that?